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Tài liệu tự học hàm số lượng giác và phương trình lượng giác - Diệp Tuân - TOANMATH.com

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<span class='text_page_counter'>(1)</span>Trung Tâm Luyện Thi Đại Học Amsterdam. 1. Chương I-Bài 1. Hàm Số Lượng Giác. HÀM SỐ LƯỢNG GIÁC-PHƯƠNG TRÌNH LƯỢNG GIÁC HÀM SỐ LƯỢNG GIÁC CƠ BẢN. §BÀI 1.. A. LÝ THUYẾT I. Ôn Tập. 1. Công thức lượng giác cơ bản. sin    tan   ,    k . cos  2 cos   cot   ,   k . sin   sin 2   cos 2   1 với mọi . k 2. . tan  .cot   1. với mọi  . . 1 cos 2  1 1  cot 2   sin 2 . với mọi   k 2. . 1  tan 2  . với mọi   k. 2. Hệ thức các cung đặc biệt Hai cung phụ nhau. . Hai cung hơn kém  :  và   .   )  sin . tan(   )  tan . cos(   )   cos . sin(   )  cos  2. cot(   )  cot . tan( )   tan . tan(   )   tan . tan(   )  cot  2. sin(   )   sin . cot( )   cot . cot(   )   cot . cot(   )  tan  2. cos(   )   cos . Hai cung đối nhau:  và . Hai cung bù nhau:  và   . cos( )  cos . sin(   )  sin . sin( )   sin . 3. Các công thức lượng giác Công Thức cộng cos(a  b)  cos a.cos b sin a.sin b.  và cos(.  2. . 2. . . . Công thức nhân đôi, ba sin 2a  2sin a cos a. Công Thức Hạ Bậc 1  cos 2a sin 2 a  2. cos 2a  cos2 a  sin 2 a 1  cos 2a sin(a  b)  sin a.cos b  cos a.sin b cos 2 a   1  2sin 2 a 2  2cos 2 a  1 tan a  tan b 1  cos 2a sin 3a  3sin a  4sin 3 a tan(a  b)  tan 2 a  3 1 tan a.tan b 1  cos 2a cos3a  4cos a  3cos a Công thức biến đổi tích thành tổng Công thức biến đổi tổng thành tích 1 ab a b cos a.cos b  [cos(a  b)  cos(a  b)] cos a  cos b  2cos .cos 2 2 2 1 ab a b sin a.sin b  [cos(a  b)  cos(a  b)] cos a  cos b  2sin .sin 2 2 2 1 ab a b sin a.cos b  [sin(a  b)  sin(a  b)] sin a  sin b  2sin .cos 2 2 2 ab a b sin a - sin b  2cos .sin 2 2 sin(a  b) sin(a  b) tan a  tan b  tan a  tan b  cos a cos b cos a cos b 4. Đổi đơn vị.. 1. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(2)</span> Trung Tâm Luyện Thi Đại Học Amsterdam Ví dụ 1. Đổi   32o sang radian. 8 7 A. B. . . 45 45. Chương I-Bài 1. Hàm Số Lượng Giác C.. 10 . 45. D.. 11 . 45. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 2. Đổi   A. 3345'.. 3 sang độ, phút, giây. 16 B. 3045'30''.. C. 3044'30''.. D. 3040'.. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... II. Tính tuần hoàn của hàm số Định nghĩa: Hàm số y  f ( x) xác định trên tập D được gọi là hàm số tuần hoàn nếu có số T  0 sao cho với mọi x  D ta có x  T  D và f ( x  T )  f ( x) . Nếu có số T dương nhỏ nhất thỏa mãn các điều kiện trên thì hàm số đó được gọi là hàm số tuần hoàn với chu kì T . Ví dụ 3. Xét tính tuần hoàn và tìm chu kỳ của các hàm số sau 1 a). y  1  sin 2 2 x . b). y  . sin 2 x Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 2. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(3)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Nhận xét: Trong quá trình làm trắc nghiệm ta sử dụng các tính chất sau Tính chất Ví dụ minh họa 2  2  y  sin  ax  b  có chu kỳ T0  Hàm số y  sin  5 x   có chu kỳ T  . . 4 a 5 . y  cos  ax  b  có chu kỳ T0 . 2 . a. y  tan  ax  b  có chu kỳ T0 .  . a. x  Hàm số y  cos   2016  có chu kỳ T  4 . 2  1 Hàm số y  tan 3 x có chu kỳ T  . 3. y  cot  ax  b  có chu kỳ T0 .  . a. x Hàm số y  cot có chu kỳ T  3 . 3. y  f1  x  có chu kỳ T1 và y  f 2  x  có chu kỳ T2 thì hàm số y  f1  x   f 2  x  có chu kỳ T0 là bội chung nhỏ nhất của T1 và T2 .. x Hàm số y  cos 2 x  sin có chu kỳ T  4 . 2 2 Vì Hàm số y  cos 2 x có chu kì T1   . 2 2 x Hàm số y  sin có chu kì T2   4 . 1 2 2.  x   Ví dụ 4. Tìm chu kì T của hàm số y  sin   2017   2 tan  2 x   . 4 2   A. T  4 . B. T   . C. T  3 . D. T  2 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 3. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(4)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Ví dụ 5. Tìm chu kì T của hàm số y  2sin 2  3 x    sin 4 x.cos x. 6  2 A. T  4 . B. T  3 . C. T  D. T  2 . . 3 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. III. Tính chẵn lẻ của hàm số Định nghĩa: Hàm số y  f  x  được goi là hàm số chẵn nếu thỏa mãn hai điều kiện;  Tập xác định của các hàm số có tính đối xứng, nghĩa là x  D suy ra  x  D .  và f   x   f  x  , x  D . Hàm số y  f  x  được goi là hàm số lẻ nếu  Tập xác định của các hàm số có tính đối xứng, nghĩa là x  D suy ra  x  D .  và f   x    f  x  , x  D . Chú ý: Nếu hàm số f  x  vi phạm một trong hai điều kiện thì ta kết luận hàm số f  x  không chẵn, không lẻ. Để chứng minh hàm số không chẵn không lẽ ta chọn hai giá trị x1  D và   x1  D sao cho.  f   x1   f  x1    f   x1    f  x1  Ví dụ 6. Xét tính chẵn, lẻ của các hàm số sau a). y  3x 2  cos 2 x . b). y  x 2 sin x  tan x . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... 4. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(5)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 7. Hàm số nào sau đây là hàm số chẵn? A. y  2cos x . B. y  2sin x .. C. y  2sin   x  .. D. y  sin x  cos x. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... sin 2 x thì y  f  x  là 2 cos x  3 A. Hàm số chẵn. B. Hàm số lẻ. C. Không chẵn không lẻ. D. Vừa chẵn vừa lẻ. Lời giải .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 8. Xét tính chẵn lẻ của hàm số y . .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................     Ví dụ 9. Xét tính chẵn lẻ của hàm số y  f  x   cos  2 x    sin  2 x   , ta được y  f  x  là: 4 4   A. Hàm số chẵn. B. Hàm số lẻ. C. Không chẵn không lẻ. D. Vừa chẵn vừa lẻ. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 5. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(6)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 10. Cho hai hàm số f  x  . 1  3sin 2 x và g  x   sin 1  x . Kết luận nào sau đây đúng về x 3. tính chẵn lẻ của hai hàm số này? A. Hai hàm số f  x  ; g  x  là hai hàm số lẻ. B. Hàm số f  x  là hàm số chẵn; hàm số f  x  là hàm số lẻ. C. Hàm số f  x  là hàm số lẻ; hàm số g  x  là hàm số không chẵn không lẻ. D. Cả hai hàm số f  x  ; g  x  đều là hàm số không chẵn không lẻ. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 11. Xét tính chẵn lẻ của hàm số f  x   sin 2007 x  cos nx , với n  . Hàm số y  f  x  là: A. Hàm số chẵn. C. Không chẵn không lẻ.. B. Hàm số lẻ. D. Vừa chẵn vừa lẻ.. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. sin 2004 n x  2004 , với n  . Xét các biểu thức sau: cos x 1, Hàm số đã cho xác định trên D  . 2, Đồ thị hàm số đã cho có trục đối xứng. 3, Hàm số đã cho là hàm số chẵn. 4, Đồ thị hàm số đã cho có tâm đối xứng. 5, Hàm số đã cho là hàm số lẻ. 6, Hàm số đã cho là hàm số không chẵn không lẻ. Số phát biểu đúng trong sáu phát biểu trên là A. 1 . B. 2 . C. 3 . D. 4 Lời giải. Ví dụ 12. Cho hàm số f  x  . 6. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(7)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 13. Xác định tất cả các giá trị của tham số m để hàm số y  f  x   3m sin 4 x  cos 2 x là hàm chẵn. A. m  0.. B. m  1.. C. m  0.. D. m  2.. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. II. Các hàm số lượng giác 1. Hàm số y  sin x Tập xác định: D  R Tập giác trị: [1;1] , tức là 1  sin x  1 x  R Hàm số đồng biến trên mỗi khoảng (.  2.  k 2 ;.  2.  k 2 ) , nghịch biến trên mỗi khoảng.  3 (  k 2 ;  k 2 ) . 2 2 Hàm số y  sin x là hàm số lẻ nên đồ thị hàm số nhận gốc tọa độ O làm tâm đối xứng. Hàm số y  sin x là hàm số tuần hoàn với chu kì T  2 . Đồ thị hàm số y  sin x . y -. -5 2 -3. -. -2 -3. 3. 2 O. 1  2. 2. . 2. 3. 2 5. x. 2. 2. 7. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(8)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. 2. Hàm số y  cos x Tập xác định: D  R Tập giác trị: [  1;1] , tức là 1  cos x  1 x  R Hàm số y  cos x nghịch biến trên mỗi khoảng (k 2 ;   k 2 ) , đồng biến trên mỗi khoảng (  k 2 ; k 2 ) . Hàm số y  cos x là hàm số chẵn nên đồ thị hàm số nhận trục Oy làm trục đối xứng. Hàm số y  cos x là hàm số tuần hoàn với chu kì T  2 . Đồ thị hàm số y  cos x .. . Đồ thị hàm số y  cos x bằng cách tịnh tiến đồ thị hàm số y  sin x theo véc tơ v  ( ;0) . 2 y. 2. 1. -. -5 -. -2. 2. -3. -3. 3 . O. . 3. 2. 2. 5. 2. 2. x. 2. 3. Hàm số y  tan x Tập xác định : D .   \   k , k   2 . Tập giá trị: Là hàm số lẻ Là hàm số tuần hoàn với chu kì T       Hàm đồng biến trên mỗi khoảng    k ;  k  2  2  Đồ thị nhận mỗi đường thẳng x . . 2.  k , k . làm một đường tiệm cận.. Đồ thị y. . - -. -2 -5. -3. 2. 2. 2. 2. 5. 3 . 2 x. 2. 2. O. 4. Hàm số y  cot x Tập xác định : D . \ k , k . . Tập giá trị: Là hàm số lẻ Là hàm số tuần hoàn với chu kì T   Hàm nghịch biến trên mỗi khoảng  k ;   k  Đồ thị nhận mỗi đường thẳng x  k , k  Đồ thị. làm một đường tiệm cận.. y. . - -. -2. 8. -5. -3. 2. 2. Lớp Toán Thầy - Diệp Tuân. 2. 2. 5. 3 . 2. 2. 2 x. O. Tel: 0935.660.880.

<span class='text_page_counter'>(9)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. B.PHƯƠNG PHÁP GIẢI TOÁN. Dạng 1. Tập xác định và tập giá trị của hàm số. 1. Phương pháp . Để tìm tập xác định của các hàm số ta dựa vào khái niệm sau: Tập xác định của hàm số y  f  x  là D   x  f  x    . Tập xác định của các hàm số cơ bản: Hàm số y . f ( x) có nghĩa  f ( x)  0 và f ( x) tồn tại. 1 có nghĩa  f ( x)  0 và f ( x) tồn tại. f ( x) sin u ( x)  0  u ( x)  k , k . Hàm số y . cos u ( x)  0  u ( x) .  2.  k , k  .. . y  tan  f  x   xác định  f  x  xác định và f  x    k ,  k  2 y  cot  f  x   xác định  f  x  xác định và f  x   k ,  k   .. .. 1  sin x, cos x  1 .. 2. Bài tập minh họa. Bài tập 1. Tập xác định của hàm số y  A. D  C. D .    \   k 2 , k   .  6     \   k 2 , k   .  3 . sin x  cos x là  cos x  2  2cos x  1 B. D  D. D .    \    k , k   .  6     \    k , k   .  3 . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 2. Tập xác định của hàm số y . A. D . \   k 2 , k . C. D .    \ k , k   .  2 . .. 1 1 1   1  sin x cos x  1 tan  x      2    B. D  \ k , k   4. D. D . \ k , k . là  . . .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... 9. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(10)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 3. Tìm tập xác định của hàm số sau:. . 1). y  tan( x  ) 6. 2). y  cot 2 (. 2  3 x) 3. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 4. Tìm tập xác định của hàm số sau: tan 2 x  1). y   cot(3x  ) sin x  1 6. 2). y . tan 5 x sin 4 x  cos 3x. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 10. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(11)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 5. Tìm tập xác định của hàm số sau: 1  cos 3x 1  cot 2 x 1  sin 2 x  1). y  2). y  3). y  tan(2 x  ) 4). y  1  sin 3 x 1  sin 4 x cos 3x  1 4 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Bài tập vận dụng. Bài 1. Tìm tập xác định của hàm số sau: 1 1). y  sin 2 x  cos 3x cot x 3). y  2sin x  1. 2). y . tan 2 x 3 sin 2 x  cos 2 x. . . 4). y  tan( x  ).cot( x  ) 4 3. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 11. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(12)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Bài 2. Tìm tập xác định của hàm số sau:. . 1). y  tan(2 x  ) 3. 2). y  tan 3x.cot 5 x. . 2  sin x tan 2 x tan 4 x 6). y  cos 4 x  sin 3x 3). y . sin 3x sin 8 x  sin 5 x Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. 4). y  tan 3x  cot( x  ) 3. 5). y . .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 12. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(13)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Câu hỏi trắc nghiệm Mức độ. Nhận biết Câu 1. Tìm tập xác định D của hàm số y  tan 2 x : A. D  C. D .   \   k 2 | k   . 4    \   k | k   . 4 . B. D  D. D .  \   k | k  2   \ k |k 2 4.  .   . . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 2. Tập xác định của hàm số y  tan 3x là..    A. D  R \   k , k  R  3 6  C. D  R \   k , k  R.   B. D  R \   k , k  R  2   2  ,k R D. D  R \ k  3 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 13. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(14)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... .................................................................................................................. Câu 3. Tập xác định của hàm số y   tan x là: A. D .   \   k , k   . 2 . B. D . \ k , k . C. D . \ k 2 , k . D. D .   \   k 2 , k   2 . .. .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 4. Tập xác định của hàm số y  tan  2 x   là: 3    5  5  A. \   k  , k  . B. \   k  , k  . 2  12  12    5  5  C. \   k  , k  . D. \   k  , k  2  6  6  Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 5. Tìm điều kiện xác định của hàm số y  tan x  cot x. k  A. x  , k . B. x   k , k  . 2 2 C. x  . D. x  k , k  . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 6. Tìm tập xác định của hàm số y  tan  2 x   . 3       A. D  \   k k   . B. D  \   k k   . 2 12  6        C. D  \   k k   . D. D  \   k k   2  6  12  Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 14. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(15)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... Câu 7. Điều kiện xác định của hàm số y  A. x  C. x . 5  k , k  . 12. . 6. k. . 2. 1  sin x là cos x B. x . , k .. 5  k , k . 12 2. D. x . . 2.  k , k . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 8. Tập xác định của hàm số y  tan 2 x là A. D  C. D .    \   k , k  . 2 4     \ k , k   .  2 . B. D  D. D .  \   k , k  2  \   k , k  4.  .   . . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 9. Tập xác định của hàm số y  tan 2 x là? A. D  C. D .   \   k , k   . 4     \ k , k   .  2 . B. D  D. D .   \   k ,k  2 4  \   k , k  2.  .   . . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 10. Tập xác định của hàm số y  tan x là:.   \   k , k   . C. . D. \ k ,k   2  Lời giải .......................................................................................................................................................................................................... .................................................................................................................. A.. \ 0 .. B.. .......................................................................................................................................................................................................... .................................................................................................................. Câu 11. Xét bốn mệnh đề sau:  5  (1) Hàm số  0;  có tập xác định là .  2  (2) Hàm số y  cos x có tập xác định là . (3) Hàm số y  tan x có tập xác định là D  (4) Hàm số y  cot x có tập xác định là D . 15. Lớp Toán Thầy - Diệp Tuân.   \   k k   . 2     \ k k   .  2 . Tel: 0935.660.880.

<span class='text_page_counter'>(16)</span> Trung Tâm Luyện Thi Đại Học Amsterdam Số mệnh đề đúng là A. 3 .. Chương I-Bài 1. Hàm Số Lượng Giác. B. 2 .. C. 1 .. D. 4. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 12. Tìm tập xác định D của hàm số y . 2017 . sin x. A. D  . C. D . \ k , k . .. B. D . \ 0.. D. D .   \   k , k   . 2 . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 13. Tìm tập xác định D của hàm số y . 1  sin x . cos x  1. A. D  . C. D . \ k , k . B. D . .. D. D .   \   k , k   . 2  \ k 2 , k  .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 14. Tìm tập xác định D của hàm số y . A. D  C. D .    \ k , k   .  2     \ 1  2k  , k   . 2  . 1.   sin  x   2 . .. B. D . \ k , k . .. D. D . \ 1  2k   , k . .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 15. Tìm tập xác định D của hàm số y  A. D  . C. D . 16.   \   k 2 , k   . 4 . Lớp Toán Thầy - Diệp Tuân. 1 . sin x  cos x B. D  D. D .    \    k , k   .  4    \   k , k   . 4 . Tel: 0935.660.880.

<span class='text_page_counter'>(17)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 16. Hàm số y  tan x  cot x . 1 1  không xác định trong khoảng nào trong các khoảng sin x cos x. sau đây?    A.  k 2 ;  k 2  với k  . 2     C.   k 2 ;   k 2  với k  . 2 . 3    k 2  với k  . B.    k 2 ; 2   D.   k 2 ; 2  k 2  với k  .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Mức độ. Thông Hiểu 1 Câu 17. Tìm tập xác định D của hàm số y  . sin x  cos x A. D . \ k | k . .. C. D .   \   k | k   . 4 . B. D .   \   k | k   . 2 . D. D . \ k 2 | k . . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................  k  \ k   là tập xác định của hàm số nào sau đây?  2  A. y  cot x . B. y  cot 2 x . C. y  tan x . D. y  tan 2 x Lời giải .......................................................................................................................................................................................................... .................................................................................................................. Câu 18. Tập D . .......................................................................................................................................................................................................... ...................................................................................................................  5 7  Câu 19. Khi x thay đổi trong khoảng  ;  thì y  sin x lấy mọi giá trị thuộc  4 4    2   2  2 A.  1;  B.   C.  1;1 . D.  ;0  ;1 .  . 2    2   2 . 17. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(18)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 20. Xét bốn mệnh đề sau: 1 : Hàm số y  sin x có tập xác định là. ..  2  : Hàm số y  cos x có tập xác định là  3 : Hàm số y  tan x có tập giá trị là .  4  : Hàm số y  cot x có tập xác định là Tìm số phát biểu đúng. A. 3 .. . .. B. 2 .. C. 4 .. D. 1. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 21. Tập xác định của hàm số y  tan x là A.. .. B.. C.. \ k , k . .. D..  \   k , k  2   \ k , k 2 2.  .   . . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 22. Tìm tập xác định D của hàm số y  A. D . \ k , k . .. C. D .   \   k , k   . 2 . tan x  1    cos  x   . sin x 3  B. D .  k  \  ,k   .  2 . D. D . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 18. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(19)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 23. Tìm tập xác định của hàm số sau y . cot x . 2sin x  1. 5     k 2 ; k   . B. D  \   k 2 , . 6  6   2     k 2 ; k   C. D  D. D  \ k ,  k 2 , . 3 3    Lời giải .......................................................................................................................................................................................................... .................................................................................................................. A. D .    \ k ,  k 2 ,   k 2 ; k  6 6   5  \ k ,  k 2 ,  k 2 ; k  6 6 . .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 24. Tìm tập xác định của hàm số y . tan x . cos x  1. A. D . \ k 2  .. B. D . C. D .   \   k ; k 2  . 2 . D. D .   \   k 2  . 2    \   k 2 ; x  k  . 2 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 25. Tìm tập xác định D của hàm số y  tan  2 x   . 4   3 k   3  ,k  . A. D  \   B. D  \   k , k   . 2 8  4   3 k    ,k  . C. D  \   D. D  \   k , k   . 2 4  2  Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 26. Tập xác định của hàm số y  tan  cos x  là: 2 . 19. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(20)</span> Trung Tâm Luyện Thi Đại Học Amsterdam A.. \ 0 .. B.. Chương I-Bài 1. Hàm Số Lượng Giác. \ 0;   .. C..   \ k  .  2. D.. \ k . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 27. Tìm tập xác định D của hàm số y  A. D  C. D .     \   k 2 ;  k 2 ; k   . 2  2     \   k 2 ; k   .  2 . 1  sin x . 1  sin x B. D . \ k ; k . .. D. D .   \   k 2 ; k   . 2 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 28. Tập xác định của hàm số y . tan 2 x là tập nào sau đây? cos x. A. D . .. B. D . C. D .    \   k  , k  . 2  4. D. D .   \   k  , k  . 2      \   k ;  k  , k  . 2 2 4 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 29. [1D1-0.0-1] Xét bốn mệnh đề sau:  5  (1) Hàm số  0;  có tập xác định là .  2  (2) Hàm số y  cos x có tập xác định là . (3) Hàm số y  tan x có tập xác định là D  (4) Hàm số y  cot x có tập xác định là D  Số mệnh đề đúng là A. 3 .. B. 2 ..   \   k k   . 2     \ k k   .  2  C. 1 .. D. 4. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 20. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(21)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 30. Tìm tập xác định D của hàm số y  A. D  C. D . tan x  5 . 1  sin 2 x. π  \   kπ, k   . 2  π  \   k 2π, k   . 2 . B. D . .. D. D . \ π  kπ, k . .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... sin 2 x  2 . 1  cos x C. D  k 2π . \ k 2π .. Câu 31. Tìm tập xác định của hàm số f  x   A. D . .. B. D . D. D . \ kπ .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 32. Tập xác định của hàm số y . tan 2 x là tập nào sau đây? cos x. A. D . .. B. D . C. D .    \   k  , k  . 2  4. D. D .   \   k  , k  . 2      \   k ;  k  , k  . 2 2 4 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 33. Tìm tập xác định của hàm số y  tan  2 x   . 3       A. D  \   k k   . B. D  \   k k   . 2 12  6        C. D  \   k k   . D. D  \   k k   2  6  12  Lời giải .......................................................................................................................................................................................................... ................................................................................................................... 21. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(22)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 34. Tìm tập xác định D của hàm số y  cot  2 x    sin 2 x. 4    A. D  \   k , k   . B. D  . 4     C. D  \   k , k   . D. D  . 2 8  Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... x  Câu 35. Tìm tập xác định D của hàm số y  3 tan 2    . 2 4    3  A. D  \   k 2 , k   . B. D  \   k 2 , k   . 2  2   3    C. D  \   k , k   . D. D  \   k , k   . 2  2  Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 36. Hàm số y . cos 2 x không xác định trong khoảng nào trong các khoảng sau đây? 1  tan x. 3    k 2  với k  . A.   k 2 ; 4 2  3  3   k 2 ;  k 2  với k  . C.  2  4 .     B.    k 2 ;  k 2  với k  . 2  2  3    k 2  với k  . D.    k 2 ; 2  . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 22. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(23)</span> Trung Tâm Luyện Thi Đại Học Amsterdam Câu 37. Tìm tập xác định D của hàm số y  A. D  C. D .   \   k 2 , k   . 2  \   k , k  .. Chương I-Bài 1. Hàm Số Lượng Giác 3 tan x  5 . 1  sin 2 x.   \   k , k   . 2  D. cos x  1  sin x  0  x  k , k  . B. D . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 38. Tìm tập xác định D của hàm số y  sin x  2. B. D   2;   .. A. D  .. C. D  0; 2 .. D. D  .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 39. Tìm tập xác định D của hàm số y  sin x  2. A. D  .. B.. \ k , k . .. C. D   1;1.. D. D  .. Lời giải.. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 40. Tìm tập xác định D của hàm số y  A. D . \ k , k . .. C. D .   \   k 2 , k   . 2 . 1 . 1  sin x B. D .   \   k , k   . 2 . D. D  .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 41. Tìm tập xác định D của hàm số y  1  sin 2 x  1  sin 2 x . A. D  . B. D  .  5  13    5   k 2  , k  .  k 2  , k  . C. D    k 2 ; D. D    k 2 ; 6 6 6   6 . 23. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(24)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Mức độ. Vận dụng.   Câu 42. Tìm tập xác định D của hàm số y  5  2 cot 2 x  sin x  cot   x  . 2   k     A. D  \  , k   . B. D  \   k , k   .  2   2  C. D  . D. D  \ k , k   .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 43. Tìm tập xác định D của hàm số y  tan  cos x  . 2    A. D  \   k , k   . B. D  2  C. D  . D. D .   \   k 2 , k   . 2  \ k , k   .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 44. Có bao nhiêu giá trị nguyên của tham số m để hàm số y  5  m sin x   m  1 cos x xác định trên A. 6 .. ? B. 8 .. C. 7 .. D. 5 .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 24. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(25)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Dạng 2. Tính chất của hàm số và đồ thị hàm số 1. Phương pháp . Trong quá trình làm trắc nghiệm ta sử dụng các tính chất sau 2 2  y  sin  ax  b  có chu kỳ T0  .  y  cos  ax  b  có chu kỳ T0  . a a  y  tan  ax  b  có chu kỳ T0 .  . a.  y  cot  ax  b  có chu kỳ T0 .  . a.  y  f1  x  có chu kỳ T1 và y  f 2  x  có chu kỳ T2 thì hàm số y  f1  x   f 2  x  có chu kỳ T0 là bội chung nhỏ nhất của T1 và T2 . Chú ý: Hàm số f ( x)  a sin ux  b cos vx  c ( với u, v  ) là hàm số tuần hoàn với chu kì T . 2 (u, v). ( (u, v) là ước chung lớn nhất). Hàm số f ( x)  a.tan ux  b.cot vx  c (với u, v  ) là hàm tuần hoàn với chu kì T .  . (u, v). 2. Bài tập vận dụng. Bài tập 6. Xét tính tuần hoàn và tìm chu kì cơ sở của các hàm số : f ( x)  cos. 3x x .cos 2 2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 7. Xét tính tuần hoàn và tìm chu kì cơ sở (nếu có) của các hàm số sau. 1). f ( x)  cos x  cos. . 3.x. . 2). f ( x)  sin x 2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. 5. Câu hỏi trắc nghiệm Mức độ. Nhận biết. 25. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(26)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. Câu 45. Trong các hàm số sau đây, hàm số nào là hàm số tuần hoàn? x 1 A. y  x  1 . B. y  x 2 . C. y  . D. y  sin x x2 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 46. Trong các hàm số sau hàm số nào tuần hoàn với chu kỳ  ? A. y  sin 2 x.. B. y  tan 2 x.. C. y  cos x.. x D. y  cot . 2. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 47. Hàm số y  cotx tuần hoàn với chu kỳ: A. T  k . B. T  2 . C. T  k 2 . D. T   . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 48. Trong các hàm số sau, hàm số nào tuần hoàn với chu kì 2 ? A. y  cos 2 x . B. y  sin x . C. y  tan x . D. y  cot x . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 49. Mệnh đề nào dưới đây sai? A. Hàm số y  tan x tuần hoàn với chu kì  . B. Hàm số y  cos x tuần hoàn với chu kì  . C. Hàm số y  cot x tuần hoàn với chu kì  . D. Hàm số y  sin 2 x tuần hoàn với chu kì  Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 50. Chu kì tuần hoàn của hàm số y  cot x là π A. . B. 2π . 2 Lời giải. 26. Lớp Toán Thầy - Diệp Tuân. C. π .. D. kπ  k . . Tel: 0935.660.880.

<span class='text_page_counter'>(27)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 51. Hàm số y  sin x tuần hoàn với chu kỳ bằng A.  . B. 2 . C.  . D. 2 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 52. Mệnh đề nào sau đây là sai? A. Hàm số y  sin x tuần hoàn với chu kì 2 . B. Hàm số y  cos x tuần hoàn với chu kì 2 . C. Hàm số y  tan x tuần hoàn với chu kì 2 . D. Hàm số y  cot x tuần hoàn với chu kì  . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 53. Trong các hàm số sau đây, hàm số nào là hàm số tuần hoàn? A. y  sin x. B. y  x  sin x. C. y  x cos x.. D y. sin x . x. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 54. Trong các hàm số sau đây, hàm số nào không tuần hoàn? A. y  cos x.. B. y  cos 2 x.. C. y  x 2 cos x .. D. y . 1 . sin 2 x. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 55. Tìm chu kì T của hàm số y  sin  5 x   . 4 . 27. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(28)</span> Trung Tâm Luyện Thi Đại Học Amsterdam A. T . 2 . 5. B. T . 5 . 2. Chương I-Bài 1. Hàm Số Lượng Giác C. T .  2. .. D. T .  8. .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... x  Câu 56. Tìm chu kì T của hàm số y  cos   2016  . 2  A. T  4 . B. T  2 . C. T  2 . D. T   . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 1 Câu 57. Tìm chu kì T của hàm số y   sin 100 x  50  . 2 1 1  A. T  . B. T  C. T  . D. T  200 2 . . 50 100 50 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Mức độ. Thông hiểu. x là số nào sau đây? 2 A. 0 . B. 2 . C. 4 . D.  . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... Câu 58. Chu kỳ của hàm số y  3sin. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 59. Tìm chu kỳ cơ sở (nếu có) của hàm số f  x   tan 2 x . A. T0  2 .. B. T0 .  2. .. C. T0   .. D. T0 .  3. .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 60. Chu kì tuần hoàn của hàm số y  sin 2 x là:  A. 3 . B. . 2 Lời giải. 28. Lớp Toán Thầy - Diệp Tuân. C. 2 .. D.  .. Tel: 0935.660.880.

<span class='text_page_counter'>(29)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 61. Trong các hàm số y  tan x ; y  sin 2 x ; y  sin x ; y  cot x , có bao nhiêu hàm số thỏa mãn tính chất f  x  k   f  x  , x  A. 3 .. , k .. B. 2 .. C. 1 .. D. 4. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 62. Hàm số y  sin 2 x có chu kỳ là  A. T  2 . B. T  . 2. C. T   .. D. T  4. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 63. Chọn khẳng định đúng trong các khẳng định sau: A. Hàm số y  tan x tuần hoàn với chu kì 2 . B. Hàm số y  cos x tuần hoàn với chu kì  ..   C. Hàm số y  sin x đồng biến trên khoảng  0;  .  2 D. Hàm số y  cot x nghịch biến trên . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 64. Mệnh đề nào sau đây đúng? A. Hàm số y  sin x tuần hoàn với chu kỳ T   ..   B. Hàm số y  sin x đồng biến trên  0;  .  2 C. Hàm số y  sin x là hàm số chẵn. D. Đồ thị hàm số y  sin x có tiệm cận ngang. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 29. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(30)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... .................................................................................................................. Câu 65. Hàm số y  sin 2 x có chu kỳ là  A. T  2 . B. T  . 2. C. T   .. D. T  4. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Mức độ. Vận dụng Câu 66. Hàm số y  cos x là hoàn tuần hoàn với chu kì là A..  . 2. B..  . 4. C. 0 .. D.  .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... x 3x Câu 67. Tìm chu kì của hàm số f  x   sin  2 cos . 2 2  A. 5 . B. . C. 4 . D. 2 2 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. x Câu 68. Tìm chu kì T của hàm số y  cos 2 x  sin . 2. A. T  4 .. B. T   .. C. T  2 .. D. T .  2. .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 69. Tìm chu kì T của hàm số y  cos 3x  cos 5 x. A. T   . B. T  3 .. 30. Lớp Toán Thầy - Diệp Tuân. C. T  2 .. D. T  5 .. Tel: 0935.660.880.

<span class='text_page_counter'>(31)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... x  Câu 70. Tìm chu kì T của hàm số y  3cos  2 x  1  2sin   3  . 2  A. T  2 . B. T  4 C. T  6 D. T   . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................     Câu 71. Tìm chu kì T của hàm số y  sin  2 x    2 cos  3 x   . 3 4   A. T  2 . B. T   . C. T  3 . D. T  4 . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 72. Tìm chu kì T của hàm số y  tan 3 x.  4 2 1 A. T  . B. T  . C. T  D. T  . . 3 3 3 3 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 73. Tìm chu kì T của hàm số y  tan 3x  cot x. A. T  4 .. B. T   .. C. T  3 .. D. T .  3. .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. x Câu 74. Tìm chu kì T của hàm số y  cot  sin 2 x. 3. 31. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(32)</span> Trung Tâm Luyện Thi Đại Học Amsterdam A. T  4 .. B. T   .. Chương I-Bài 1. Hàm Số Lượng Giác C. T  3 .. D. T .  3. .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... x   Câu 75. Tìm chu kì T của hàm số y  sin  tan  2 x   . 2 4  A. T  4 . B. T   . C. T  3 . D. T  2 . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 76. Tìm chu kì T của hàm số y  2 cos 2 x  2017. A. T  3 . B. T  2 . C. T   . D. T  4 . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 77. Tìm chu kì T của hàm số y  2sin 2 x  3cos 2 3 x. A. T   .. B. T  2 .. C. T  3 .. D. T .  3. .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 78. Tìm chu kì T của hàm số y  tan 3 x  cos 2 2 x.   A. T   . B. T  . C. T  . D. T  2 . 3 2 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 32. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(33)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 79. Hàm số nào sau đây có chu kì khác  ?     A. y  sin   2 x  . B. y  cos 2  x   . C. y  tan  2 x  1 . D. y  cos x sin x. 4 3   Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 80. Hàm số nào sau đây có chu kì khác 2 ? x x x  A. y  cos3 x. B. y  sin cos . C. y  sin 2  x  2  . D. y  cos 2   1 . 2 2 2  Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 81. Hai hàm số nào sau đây có chu kì khác nhau? x A. y  cos x và y  cot . B. y  sin x và y  tan 2 x. 2 x x C. y  sin và y  cos . D. y  tan 2 x và y  cot 2 x. 2 2 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 82. Một vật nặng treo bởi một chiếc lò xo, chuyển động lên xuống qua vị trí cân bằng (hình vẽ). Khoảng cách h từ vật đến vị trí cân bằng ở thời điểm t giây được tính theo công thức h  d trong đó d  5sin 6t  4cos6t với d được tính bằng centimet.. 33. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(34)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. Ta quy ước rằng d  0 khi vật ở trên vị trí cân bằng, d  0 khi vật ở dưới vị trí cân bằng. Hỏi trong giây đầu tiên, có bao nhiêu thời điểm vật ở xa vị trí cân bằng nhất? A. 0 . B. 4 . C. 1 . D. 2 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 83. Trong các hàm số y  tan x ; y  sin 2 x ; y  sin x ; y  cot x , có bao nhiêu hàm số thỏa mãn tính chất f  x  k   f  x  , x  A. 3 .. B. 2 .. , k . C. 1 .. D. 4 .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 34. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(35)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. Dạng 3. Tính chẵn, lẻ của hàm số 1. Định nghĩa: Hàm số y  f  x  được goi là hàm số chẵn nếu thỏa mãn hai điều kiện;  Tập xác định của các hàm số có tính đối xứng, nghĩa là x  D suy ra  x  D .  và f   x   f  x  , x  D . Hàm số y  f  x  được goi là hàm số lẻ nếu  Tập xác định của các hàm số có tính đối xứng, nghĩa là x  D suy ra  x  D .  và f   x    f  x  , x  D . Chú ý: Nếu hàm số f  x  vi phạm một trong hai điều kiện thì ta kết luận hàm số f  x  không chẵn, không lẻ. Để chứng minh hàm số không chẵn không lẽ ta chọn hai giá trị x1  D và   x1  D sao cho.  f   x1   f  x1    f   x1    f  x1 . 2. Bài tập vận dụng. Bài tập 8. Xét tính chẵn, lẻ của các hàm số sau 1   a). y  5cos  2 x   . b). y   cos 2 x . 3 x  1   Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. Bài tập 9. Xét tính chẵn, lẻ của các hàm số sau cos3 x  sin 2 x sin x  tan x a). y  . b). y  . cos 2 x sin x  cot x Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 35. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(36)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................ 1. Câu hỏi trắc nghiệm. Mức độ. Nhận biết Câu 84. Trong các hàm số sau, hàm số nào là hàm số chẵn? A. y  sin x. B. y  cos x. C. y  tan x. D. y  cot x. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 85. Trong các hàm số sau, hàm số nào là hàm số chẵn? A. y   sin x. B. y  cos x  sin x. C. y  cos x  sin 2 x. D. y  cos x sin x. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. 36. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(37)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................ Câu 86. Trong các hàm số sau, hàm số nào là hàm số chẵn? A. y  sin 2 x.. B. y  x cos x.. C. y  cos x.cot x.. D. y . tan x . sin x. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................. .......................................................................................................................................................................................................... ................................................................................................................ Câu 87. Trong các hàm số sau, hàm số nào là hàm số chẵn? A. y  sin x .. B. y  x 2 sin x.. C. y . x . cos x. D. y  x  sin x.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Mức độ 2. Thông Hiểu Câu 88. Trong các hàm số sau, hàm số nào có đồ thị đối xứng qua trục tung? tan x   . A. y  sin x cos 2 x. B. y  sin 3 x.cos  x   . C. y  D. y  cos x sin 3 x. 2 2 tan x  1  Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 89. Trong các hàm số sau, hàm số nào là hàm số lẻ? A. y  cos x  sin 2 x. B. y  sin x  cos x. C. y   cos x. D. y  sin x.cos 3x. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 37. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(38)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. Câu 90. Trong các hàm số sau, hàm số nào có đồ thị đối xứng qua gốc tọa độ? sin x  1 A. y  cot 4 x. B. y  C. y  tan 2 x. D. y  cot x . . cos x Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 91. Trong các hàm số sau, hàm số nào là hàm số lẻ? cot x tan x   A. y  sin   x  . B. y  sin 2 x. C. y  D. y  . . cos x sin x 2  Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 92. Trong các hàm số sau, hàm số nào là hàm số lẻ? A. y  1  sin 2 x. B. y  cot x .sin 2 x. C. y  x 2 tan 2 x  cot x.. D. y  1  cot x  tan x .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Mức độ 3. Vận dụng Câu 93. Cho hàm số f  x   sin 2 x và g  x   tan 2 x. Chọn mệnh đề đúng A. f  x  là hàm số chẵn, g  x  là hàm số lẻ. B. f  x  là hàm số lẻ, g  x  là hàm số chẵn. C. f  x  là hàm số chẵn, g  x  là hàm số chẵn. D. f  x  và g  x  đều là hàm số lẻ.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 38. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(39)</span> Trung Tâm Luyện Thi Đại Học Amsterdam Câu 94. Cho hai hàm số f  x   A. f  x  lẻ và g  x  chẵn.. Chương I-Bài 1. Hàm Số Lượng Giác. sin 2 x  cos 3 x cos 2 x và g  x   . Mệnh đề nào sau là đúng? 2 2  tan 2 x 1  sin 3x B. f  x  và g  x  chẵn.. C. f  x  chẵn, g  x  lẻ.. D. f  x  và g  x  lẻ. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 95. Trong các hàm số sau, hàm số nào có đồ thị đối xứng qua gốc tọa độ? 1     A. y  3 . B. y  sin  x   . C. y  2 cos  x   . D. y  sin 2 x . 4 4 sin x   Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 96. Mệnh đề nào sau đây là sai? A. Đồ thị hàm số y  sin x đối xứng qua gốc tọa độ O. B. Đồ thị hàm số y  cos x đối xứng qua trục Oy. C. Đồ thị hàm số y  tan x đối xứng qua trục Oy.. D. Đồ thị hàm số y  tan x đối xứng qua gốc tọa độ O. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 97. Trong các hàm số sau, hàm số nào là hàm số chẵn?       A. y  2 cos  x    sin   2 x  . B. y  sin  x    sin  x   . 2 4 4      C. y  2 sin  x    sin x. D. y  sin x  cos x . 4 . 39. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(40)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 98. Trong các hàm số sau, hàm số nào là hàm số lẻ ?     A. y  x 4  cos  x   . B. y  x 2017  cos  x   . 2 3   C. y  2015  cos x  sin 2018 x. D. y  tan 2017 x  sin 2018 x. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Dạng 4. Giá trị lớn nhất và giá trị nhỏ nhất 1. Phương pháp chung. Số M được gọi là giá trị lớn nhất của hàm số f  x  trên X nếu. x  X : f  x   M . Kí hiệu: M  max f  x  .  X x0  X : f  x0   M Số m được gọi là giá trị nhỏ nhất của hàm số f  x  trên X nếu. x  X : f  x   m . Kí hiệu: m  min f  x  .  X  x  X : f x  m    0 0 Trắc nghiệm: Tìm GTLN và GTNN của một hàm số y  f  x  trên  a ; b  . Bước 1. Nhấn MODE 7 (TABLE) Bước 2. Nhập biểu thức f  x  vào máy Bước 3. Nhấn = sau đó nhập Start  a , End  b , Step . b-a . (Có thể lấy từ 29 trở xuống) 20. (Chia 20 để có được 20 bước nhảy, và bảng TABLE có 21 giá trị, như thế là đủ!) Bước 4. Sau đó, dựa vào bảng TABLE, ta tìm GTNN và GTLN.. 40. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(41)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. 2. Các trường hợp. Trường hợp 1. Sử dụng miền giá trị để suy ra giá trị lớn nhất và giá trị nhỏ nhất Nếu ta biến đổi hàm số f  x  về dạng m  f  x   M thì M  max f  x  , m  min f  x  . Để làm X. được điều đó ta sử dụng các tính chất sau:  1  sin f  x   1.  1  cos f  x   1. 2  0  sin f  x   1. 2  0  cos f  x   1.  0  sin f  x   1.  0  cos f  x   1. X. 3. Bài tập minh họa.. Bài tập 10. Giá trị nhỏ nhất và giá trị lớn nhất của hàm số y  3  2sin 2 x lần lượt là A. 3 ; 0. B. 0 ; 1. C. 1 ; 3. D. 1 ; 2. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 11. Tìm giá trị lớn nhất và giá trị nhỏ nhất của các hàm số sau   a). y  3  sin  2 x   . b). y  5  4sin 2 x cos 2 x . c). y  1  sin  x 2   1 . 4    d). y  tan x  cot x . e). y  4sin 2 x  2 sin  2 x   f). y  sin 6 x  cos 6 x 4  Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 41. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(42)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Bài tập 12. Tìm giá trị lớn nhất và giá trị nhỏ nhất của các hàm số sau   a). y  2sin x  3 . b). y  1  3sin  2 x   . 4  d). y  1  2  sin 2 x .. e). y  1  2 cos 2 x  1. c). y  3  2 cos 2 3 x . f). y . 4 1  2sin 2 x. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 42. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(43)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Bài tập 13. Tìm tập giá trị lớn nhất, giá trị nhỏ nhất của các hàm số sau. a). y  4sin x cos x  1 b). y  4  3sin 2 2 x c). y  2sin 3x  1 d). y  3  4 cos 2 2 x. e). y  1  2 4  cos3x f). x  D Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(44)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 14. Tìm giá trị lớn nhất và giá trị nhỏ nhất của các hàm số sau   2  a). y  sin x trên đoạn   ;  .  3 3         b). y  cos  2 x    cos  2 x   trên đoạn   ;  4 4    3 6 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Câu hỏi trắc nghiệm. Mức độ. Nhận biết Câu 99. Tìm giá trị lớn nhất M và giá trị nhỏ nhất m của hàm số y  3sin x  2. A. M  1, m  5. B. M  3, m  1. C. M  2, m  2. D. M  0, m  2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 44. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(45)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 100. Tìm tập giá trị T của hàm số y  3cos 2 x  5. A. T   1;1. B. T   1;11. C. T   2;8.. D. T  5;8.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 101. Tìm tập giá trị T của hàm số y  5  3sin x. A. T   1;1. B. T   3;3.. C. T   2;8.. D. T  5;8.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 102. Cho hàm số y  2sin  x    2 . Mệnh đề nào sau đây là đúng? 3  A. y  4, x  . B. y  4, x  . C. y  0, x  . D. y  2, x  . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Mức độ 2. Thông hiểu Câu 103. Hàm số y  5  4sin 2 x cos 2 x có tất cả bao nhiêu giá trị nguyên? A. 3. B. 4. C. 5. D. 6. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 104. Tìm giá trị nhỏ nhất m của hàm số y   2 sin  2016 x  2017  . A. m  2016 2.. B. m   2.. C. m  1. D. m  2017 2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 45. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(46)</span> Trung Tâm Luyện Thi Đại Học Amsterdam Câu 105. Tìm giá trị nhỏ nhất m của hàm số y . 1 A. m  . 2. B. m . Chương I-Bài 1. Hàm Số Lượng Giác 1 . cos x  1. 1 . 2. C. m  1.. D. m  2.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 106. Gọi M , m lần lượt là giá trị lớn nhất và giá trị nhỏ nhất của hàm số y  sin x  cos x . Tính P  M  m. A. P  4. B. P  2 2. C. P  2. D. P  2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 107. Tập giá trị T của hàm số y  sin 2017 x  cos 2017 x. A. T   2; 2.. B. T   4034; 4034.. C. T   2; 2  .  . D. T  0; 2  .  . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 108. Hàm số y  sin  x    sin x có tất cả bao nhiêu giá trị nguyên? 3  A. 1. B. 2. C. 3. D. 4. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 46. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(47)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. Câu 109. Hàm số y  sin 4 x  cos 4 x đạt giá trị nhỏ nhất tại x  x0 . Mệnh đề nào sau đây là đúng? A. x0  k 2 , k  .. B. x0  k , k  .. C. x0    k 2 , k  .. D. x0 .  2.  k , k  .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 110. Tìm giá trị lớn nhất M và giá trị nhỏ nhất m của hàm số y  1  2 cos3x . A. M  3, m  1.. B. M  1, m  1. C. M  2, m  2. D. M  0, m  2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 111. Tìm giá trị lớn nhất và giá trị nhỏ nhất của hàm số y  7  3cos 2 x A. M  10, m  2.. B. M  7, m  2.. C. M  10, m  7.. D. M  0, m  1.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 112. Hàm số y  1  2 cos 2 x đạt giá trị nhỏ nhất tại x  x0 . Mệnh đề nào sau đây là đúng? A. x0    k 2 , k  .. B. x0 .  2.  k , k  .. C. x0  k 2 , k  .. D. x0  k , k  .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 113. Tìm giá trị lớn nhất M của hàm số y  4sin 2 x  2 sin  2 x   . 4  A. M  2. B. M  2  1. C. M  2  1. D. M  2  2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... 47. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(48)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 114. Tìm tập giá trị T của hàm số y  sin 6 x  cos 6 x. 1  1   1 A. T   0; 2. B. T   ;1 . C. T   ;1 . D. T  0;  . 2  4   4 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 115. Cho hàm số y  cos 4 x  sin 4 x . Mệnh đề nào sau đây là đúng? A. y  2, x  .. B. y  1, x  .. C. y  2, x  .. D. y . 2 , x  . 2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 116. Tìm giá trị lớn nhất M và nhỏ nhất m của hàm số y  sin 2 x  2 cos 2 x. A. M  3, m  0. B. M  2, m  0. C. M  2, m  1. D. M  3, m  1. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 117. Tìm giá trị lớn nhất M của hàm số y . 1 A. M  . 2. 2 B. M  . 3. 2 . 1  tan 2 x C. M  1.. D. M  2.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... 48. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(49)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 118. Gọi M , m lần lượt là giá trị lớn nhất và giá trị nhỏ nhất của hàm số y  8sin 2 x  3cos 2 x . Tính P  2M  m2 . A. P  1. B. P  2. C. P  112. D. P  130. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 119. Số giờ có ánh sáng mặt trời của một thành phố A trong ngày thứ của năm được cho bởi một hàm số với và . Vào ngày nào trong năm thì thành phố A có nhiều giờ có ánh sáng mặt trời nhất? A. 28 tháng 5. B. 29 tháng 5. C. 30 tháng 5. D. 31 tháng 5. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 120. Hằng ngày mực nước của con kênh lên xuống theo thủy triều. Độ sâu (mét) của mực nước trong kênh được tính tại thời điểm (giờ) trong một ngày bởi công thức Mực nước của kênh cao nhất khi: A. (giờ). B. (giờ). C. (giờ). D. t  16 (giờ) Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 49. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(50)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. Trường hợp 2. Sử dụng tính chất hình học( đồ thị Parabol của hàm bậc 2). 1. Phương pháp. Nếu ta biến đổi hàm số f  x  về dạng m  f  x   M thì M  max f  x  , m  min f  x  . Để làm X. X. được điều đó ta sử dụng các tính chất sau: Đặt t  sin f  x  hoặc t  cos f  x  thì 1  t  1. 2 Hàm số bậc hai y  ax  bx  c  a  0  xác định trên tập R. Nếu hệ số a  0. Nếu hệ số a  0. 2. Bài tập minh họa.. 2    Bài tập 15. Tập giá trị của hàm số y  2sin 2 x  sin x  4 với x    ; là 3   6  30   30   31  A.  4 ; 7. B.  C.  D.  ; 7  . ; 7 . ; 4 . 8  8  8  Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 16. Tìm giá trị lớn nhất và giá trị nhỏ nhất của các hàm số sau a). y  cos 2 x  2sin x  2 . b). y  sin 4 x  2 cos 2 x  1 . c). y  3sin 4 x  cos 4 x . d). y  2sin 4 x  cos 4 x . e). y  2sin 2 x  cos 2 2 x f). y  tan 2 x  4 tan x  1 g). y  tan 2 x  cot 2 x  3(tan x  cot x)  1. 50. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(51)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(52)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 17. Tìm tập giá trị lớn nhất, giá trị nhỏ nhất của các hàm số sau. 1). y  6 cos 2 x  cos 2 2 x 2). y  (4sin x  3cos x) 2  4(4sin x  3cos x)  1 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 18. Tìm tất cả các giá trị của tham số m để hàm số sau chỉ nhận giá trị dương : y  (3sin x  4 cos x) 2  6sin x  8cos x  2m  1 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 52. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(53)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. Bài tập 19. Tìm m để hàm số y  2sin 2 x  4sin x cos x  (3  2m) cos 2 x  2 xác định với mọi x .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. 3. Câu hỏi trắc nghiệm Mức độ 3. Vận dụng. Câu 121. Gọi M , m lần lượt là giá trị lớn nhất và giá trị nhỏ nhất của hàm số y  sin 2 x  4sin x  5 . Tính P  M  2m2 . A. P  1. B. P  7. C. P  8. D. P  2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 122. Hàm số y  cos 2 x  cos x có tất cả bao nhiêu giá trị nguyên? A. 1. B. 2. C. 3. D. 4. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 123. Hàm số y  cos 2 x  2sin x  2 đạt giá trị nhỏ nhất tại x . Mệnh đề nào sau đây là đúng? A. x0 .  2.  k 2 , k  .. C. x0    k 2 , k  .. .  k 2 , k  . 2 D. x0  k 2 , k  . B. x0  . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 53. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(54)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 124. Tìm giá trị lớn nhất và nhất của hàm số A. M  2, m  2. B. M  1, m  0. C. M  4, m  1. D. M  2, m  1. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 125. Tìm giá trị nhỏ nhất của hàm số y  4sin 4 x  cos 4 x . A. m  3. B. m  1. C. m  3. D. m  5. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Trường hợp 3. Sử dụng tính chất của hàm số a cos f  x   b sin f  x   c 1 1. Phương pháp. Áp dụng điều kiện có nghĩa của phương trình a cos f  x   b sin f  x   c là a 2  b 2  c 2 2. Bài tập minh họa. Bài tập 20. Gọi M là giá trị lớn nhất và m là giá trị nhỏ nhất của hàm số y  Khi đó M 2  m2 bằng 5 A. . 3. B.. 2 3 . 3. C.. 4 . 3. 1  sin x . 2  cos x. D.. 16 . 9. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 54. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(55)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 21. Tìm giá trị lớn nhất và giá trị nhỏ nhất của các hàm số sau a). y  sin x  3 cos x  3 b). y  2sin 2 x  3sin x cos x  5cos 2 x . sin x  2 cos x  1 c). y  d). y  3sin x  4cos x  1 sin x  cos x  2 e). y  2sin 2 x  3sin 2 x  4 cos 2 x f). y  sin 2 x  3sin 2 x  3cos 2 x Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(56)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 22. Tìm gtln và gtnn của các hàm sau : a). y  3sin x  4cos x  5 c). y . 3sin 2 x  cos 2 x sin 2 x  4 cos 2 x  1. e). y  3 cos x  sin x  4. b). y . sin x  2 cos x  1 sin x  cos x  2. d). y  4sin 3x  3cos3x  1. sin 2 x  2cos 2 x  3 2sin 2 x  cos 2 x  4 sin 2 2 x  3sin 4 x h). y  2 cos 2 2 x  sin 4 x  2 f). y . 2sin 2 3 x  4sin 3 x cos 3 x  1 sin 6 x  4 cos 6 x  10 k). y  3(3sin x  4 cos x) 2  4(3sin x  4 cos x)  1 Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. g). y . .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 56. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(57)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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.......................................................................................................................................................................................................... ................................................................................................................... Bài tập 23. Tìm m để hàm số xác định với mọi x và bất phương trình sau đúng với mọi x  a). y  5sin 4 x  6cos 4 x  2m  1 b). (3sin x  4 cos x) 2  6sin x  8cos x  2m  1 3sin 2 x  cos 2 x 4sin 2 x  cos 2 x  17 c). d).  m 1 2 2 sin 2 x  4 cos x  1 3cos 2 x  sin 2 x  m  1 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 57. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(58)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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.......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 24. Tìm k để giá trị nhỏ nhất của hàm số y . k sin x  1 lớn hơn 1 . cos x  2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. 4. Câu hỏi trắc nghiệm Mức độ. Nhận biết Câu 126. Tìm giá trị nhỏ nhất m của hàm số y  2sin 2 x  3 sin 2 x . A. m  2  3.. B. m  1.. C. m  1.. D. m   3.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 58. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(59)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 127. Tìm tập giá trị T của hàm số y  12sin x  5cos x. A. T   1;1.. B. T   7;7.. C. T   13;13.. D. T   17;17.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 128. Tìm giá trị lớn nhất M của hàm số y  4sin 2 x  3cos 2 x. A. M  3. B. M  1. C. M  5. D. M  4. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Dạng 5. Khảo sát sự biến thiên và vẽ đồ thị của hàm số 1. Bài tập minh họa Bài tập 25. Xét sự biến thiên và vẽ đồ thị hàm số sau y  2sin x Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 59. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(60)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập25. Xét sự biến thiên và vẽ đồ thị hàm số sau y  tan 2 x Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 26. Xét sự biến thiên và vẽ đồ thị hàm số sau y  1  2 cos 2 x Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................  31 33  Bài toán 27. Với x   ;  , mệnh đề nào sau đây là đúng? 4   4 A. Hàm số y  cos x nghịch biến. B. Hàm số y  sin x đồng biến. C. Hàm số y  tan x nghịch biến. D. Hàm số y  cot x nghịch biến. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... 60. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(61)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. 5. Câu hỏi trắc nghiệm. Mức độ. Nhận biết Câu 129. Cho hàm số y  sin x . Mệnh đề nào sau đây là đúng?    3  A. Hàm số đồng biến trên khoảng  ;   , nghịch biến trên khoảng   ;  . 2  2    3      B. Hàm số đồng biến trên khoảng   ;   , nghịch biến trên khoảng   ;  . 2  2  2 2      C. Hàm số đồng biến trên khoảng  0;  , nghịch biến trên khoảng   ; 0  .  2  2       3  D. Hàm số đồng biến trên khoảng   ;  , nghịch biến trên khoảng  ;  .  2 2 2 2  Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................  31 33  ; Câu 130. Với x    , mệnh đề nào sau đây là đúng? 4   4 A. Hàm số y  cot x nghịch biến. B. Hàm số y  tan x nghịch biến. C. Hàm số y  sin x đồng biến. D. Hàm số y  cos x nghịch biến. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 131. Với x   0;  , mệnh đề nào sau đây là đúng?  4 A. Cả hai hàm số y   sin 2 x và y  1  cos 2 x đều nghịch biến. B. Cả hai hàm số y   sin 2 x và y  1  cos 2 x đều đồng biến. C. Hàm số y   sin 2 x nghịch biến, hàm số y  1  cos 2 x đồng biến. D. Hàm số y   sin 2 x đồng biến, hàm số y  1  cos 2 x nghịch biến. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 61. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(62)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... Câu 132. Hàm số y  sin 2 x đồng biến trên khoảng nào trong các khoảng sau?      3   3  A.  0;  . B.  ;   . C.   ;  . D.  ; 2  2   4 2    2  Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................    Câu 133. Trong các hàm số sau, hàm số nào đồng biến trên khoảng   ;  ?  3 6         A. y  tan  2 x   . B. y  cot  2 x   . C. y  sin  2 x   . D. y  cos  2 x   6 6 6 6     Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 134. Đồ thị hàm số y  cos  x   được suy từ đồ thị  C  của hàm số y  cos x bằng cách: 2  A. Tịnh tiến  C  qua trái một đoạn có độ dài là B. Tịnh tiến  C  qua phải một đoạn có độ dài là C. Tịnh tiến  C  lên trên một đoạn có độ dài là. . 2. ..  2. . 2. .. .. D. Tịnh tiến  C  xuống dưới một đoạn có độ dài là.  2. .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 62. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(63)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. Câu 135. Đồ thị hàm số y  sin x được suy từ đồ thị  C  của hàm số y  cos x bằng cách: A. Tịnh tiến  C  qua trái một đoạn có độ dài là B. Tịnh tiến  C  qua phải một đoạn có độ dài là C. Tịnh tiến  C  lên trên một đoạn có độ dài là.  2. ..  2. . 2. .. .. D. Tịnh tiến  C  xuống dưới một đoạn có độ dài là.  2. .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 136. Đồ thị hàm số y  sin x được suy từ đồ thị  C  của hàm số y  cos x  1 bằng cách:.  và lên trên 1 đơn vị. 2  B. Tịnh tiến  C  qua phải một đoạn có độ dài là và lên trên 1 đơn vị. 2  C. Tịnh tiến  C  qua trái một đoạn có độ dài là và xuống dưới 1 đơn vị. 2  D. Tịnh tiến  C  qua phải một đoạn có độ dài là và xuống dưới 1 đơn vị. 2 Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. A. Tịnh tiến  C  qua trái một đoạn có độ dài là. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 137. Đường cong trong hình dưới đây là đồ thị của một hàm số trong bốn hàm số được liệt kê ở bốn phương án A, B, C, D.. Hỏi hàm số đó là hàm số nào? A. y  1  sin 2 x. B. y  cos x.. C. y   sin x.. D. y   cos x.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 63. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(64)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... .................................................................................................................. Câu 138. Đường cong trong hình dưới đây là đồ thị của một hàm số trong bốn hàm số được liệt kê ở bốn phương án A, B, C, D.. Hỏi hàm số đó là hàm số nào? x x A. y  sin . B. y  cos . 2 2. x C. y   cos . 4.  x D. y  sin    .  2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 139. Đường cong trong hình dưới đây là đồ thị của một hàm số trong bốn hàm số được liệt kê ở bốn phương án A, B, C, D.. Hỏi hàm số đó là hàm số nào? 2x 2x A. y  cos . B. y  sin . 3 3. C. y  cos. 3x . 2. D. y  sin. 3x . 2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 140. Đường cong trong hình dưới đây là đồ thị của một hàm số trong bốn hàm số được liệt kê ở bốn phương án A, B, C, D.. Hỏi hàm số đó là hàm số nào?  3        A. y  sin  x   . B. y  cos  x  C. y  2 sin  x   . D. y  cos  x   . . 4 4  4 4     Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 64. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(65)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 141. Đường cong trong hình dưới đây là đồ thị của một hàm số trong bốn hàm số được liệt kê ở bốn phương án A, B, C, D.. Hỏi hàm số đó là hàm số nào?   A. y  sin  x   . 4    C. y  2 sin  x   . 4 .   B. y  cos  x   . 4    D. y  2 cos  x   . 4 . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 142. Đường cong trong hình dưới đây là đồ thị của một hàm số trong bốn hàm số được liệt kê ở bốn phương án A, B, C, D.. Hỏi hàm số đó là hàm số nào? A. y  sin x. B. y  sin x .. C. y  sin x .. D. y   sin x.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 143. Đường cong trong hình dưới đây là đồ thị của một hàm số trong bốn hàm số được liệt kê ở bốn phương án A, B, C, D.. Hỏi hàm số đó là hàm số nào? A. y  cos x. B. y   cos x. C. y  cos x .. D. y  cos x .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 65. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(66)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 144. Đường cong trong hình dưới đây là đồ thị của một hàm số trong bốn hàm số được liệt kê ở bốn phương án A, B, C, D.. Hỏi hàm số đó là hàm số nào? A. y  sin x . B. y  sin x .. C. y  cos x .. D. y  cos x .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 145. Đường cong trong hình dưới đây là đồ thị của một hàm số trong bốn hàm số được liệt kê ở bốn phương án A, B, C, D.. Hỏi hàm số đó là hàm số nào? A. y  tan x. B. y  cot x.. C. y  tan x .. D. y  cot x .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 146. Đường cong trong hình dưới đây là đồ thị của một hàm số trong bốn hàm số được liệt kê ở bốn phương án A, B, C, D.. Hỏi hàm số đó là hàm số nào?   A. y  sin  x    1. 2    C. y   sin  x    1. 2 .   B. y  2sin  x   . 2    D. y  sin  x    1. 2 . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 66. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(67)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 1. Hàm Số Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 147. Đường cong trong hình dưới đây là đồ thị của một hàm số trong bốn hàm số được liệt kê ở bốn phương án A, B, C, D.. Hỏi hàm số đó là hàm số nào? A. y  1  sin x . B. y  sin x .. C. y  1  cos x .. D. y  1  sin x. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 148. Đường cong trong hình dưới đây là đồ thị của một hàm số trong bốn hàm số được liệt kê ở bốn phương án A, B, C, D.. Hỏi hàm số đó là hàm số nào? A. y  1  sin x . B. y  sin x .. C. y  1  cos x .. D. y  1  sin x. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 67. Lớp Toán Thầy - Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(68)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. PHƯƠNG TRÌNH LƯỢNG GIÁC CƠ BẢN. §BÀI 2. A. LÍ THUYẾT. I. Phương trình: sin x  m (1) 1. Phương pháp. Nếu: m  1  Phương trình vô nghiệm, vì 1  sin x  1 với mọi x ..  1 2 3     ; ; 1 . Nếu: m  1      ;  và m  0;  ;  2 2 2  2 2   Đặt sin   m  x    k 2  (1)  sin x  sin     x      k 2. ( k  ).. 2. Chú ý :.    1 2 3       ; ; 1 thì Nếu  thỏa mãn  2 2 và m  0;  ;  2 2 2  sin   m   x  arcsin m  k 2 sin x  m   , k  x    arcsin m  k 2 Các trường hợp đặc biệt:. sin x  1  x . . 2. ..  k 2 .. .  k 2 . 2 sin x  0  x  k . 3. Ví dụ minh họa. Ví dụ 1. Giải các phương trình sau . 1 1   1   a). sin  2 x     . b). sin(4 x  )  . c). 2sin  2 x    3  0. 4 3 2 2 3           d). 3sin  4 x    4  0 . e). sin  2 x    sin  x   . f). sin  x    0 . 3 2 4 4     Lời giải .......................................................................................................................................................................................................... .................................................................................................................. sin x  1  x  . .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 68. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(69)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Câu hỏi trắc nghiệm Mức độ 2. Thông hiểu  2x   Câu 1. Giải phương trình sin     0 .  3 3. A. x  k  k  C. x .  3. ..  k  k . 2 k 3   k  . 3 2  k 3 D. x    k  . 2 2 B. x . .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 69. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(70)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản.  1  Câu 2. Số vị trí biểu diễn các nghiệm của phương trình sin  2 x    trên đường tròn lượng 3 2  giác là? A. 1. B. 2. C. 4. D. 6. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Câu 3. Với những giá trị nào của x thì giá trị của các hàm số y  sin 3x và y  sin x bằng nhau?  x  k 2  x  k  A. B.   k  .  k  .  x    k 2 x    k   4  4 2 C. x  k. . 4. k  .. D. x  k. . 2. k  .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Mức độ 3. Vận dụng Câu 4. Gọi x0 là nghiệm dương nhỏ nhất của phương trình đúng?.   A. x0   0;  .  4. 70.    B. x0   ;  . 4 2 Lời giải.. Lớp Toán Thầy-Diệp Tuân. 2 cos 2 x  0 . Mệnh đề nào sau đây là 1  sin 2 x.   3 C. x0   ; 2 4.  . .  3  D. x0   ;   .  4 . Tel: 0935.660.880.

<span class='text_page_counter'>(71)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... . . Câu 5. Hỏi trên đoạn  2017; 2017 , phương trình  sin x  1 sin x  2  0 có tất cả bao nhiêu nghiệm? A. 4034.. B. 4035.. C. 641.. D. 642.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................  3  Câu 6. Tổng nghiệm âm lớn nhất và nghiệm dương nhỏ nhất của sin  3x    bằng: 4 2  A..  . 9. B. .  . 6. C..  . 6. D. .  . 9. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... ......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 71. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(72)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản.   Câu 7. Tìm nghiệm dương nhỏ nhất của phương trình 2sin  4 x    1  0. 3   7   A. x  . B. x  C. x  . D. x  . . 4 24 8 12 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. II. Phương trình: cos x  m (2) 1. Phương pháp. Nếu: m  1  phương trình vô nghiệm vì 1  cos x  1 với mọi x .. 1 2 3   ; ; 1 ta đặt : cos   m Nếu: m  1    [0;  ] và m  0;  ;  2 2 2    x    k 2 ( k  Z ). (2)  cos x  cos     x    k 2 2. Chú ý ..   0     1 2 3 ; ; 1 thì Nếu  thỏa mãn  và m  0;  ;  2 2 2  cos   m   x  arc cos m  k 2 cos x  m   , k .  x   arc cos m  k 2 Các trường hợp đặc biệt:. cos x  1  x  k 2 cos x  1  x    k 2  cos x  0  x   k 2 3. Ví dụ minh họa. Ví dụ 2. Giải các phương trình sau . a). cos  3 x  150  . 3 2.   d). cos  5 x    1  0 3 . b). 2 cos x  2  0.   c). cos  3 x    0 4    f). 2 cos  2 x    3  0 6 .   e). 2 cos  3 x    1  0 4  Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 72. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(73)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Câu hỏi trắc nghiệm Mức độ 2. Thông Hiểu Câu 8. Gọi x0 là nghiệm âm lớn nhất của phương trình cos  5 x  450   là đúng? A. x0   300 ;00  .. B. x0   450 ; 300  .. 3 . Mệnh đề nào sau đây 2. C. x0   600 ; 450  .. D. x0   900 ; 600 . .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 73. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(74)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. Câu 9. Gọi S là tập nghiệm của phương trình 2 cos x  3  0 . Khẳng định nào sau đây là đúng? 5 11 13 13 A. B. C. D.   S.  S.  S.  S. 6 6 6 6 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 7 là một nghiệm của phương trình nào sau đây? 3 A. 2sin x  3  0. B. 2sin x  3  0. C. 2 cos x  3  0. D. 2 cos x  3  0. Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. Câu 10. Hỏi x . .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Mức độ 3. Vận dụng. 13    Câu 11. Hỏi trên đoạn   ; 2  , phương trình cos x  có bao nhiêu nghiệm? 14  2  A. 2 . B. 3 . C. 4 . D. 5 . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 74. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(75)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. III. Phương trình : tan x  m (3) 1. Phương pháp.. .  k  k   . 2 1      ; 1;  3  . Ta đặt tan   m Với m      ;  và m  0;  3  2 2   (3)  tan x  tan   x    k , k  . Điều kiện: x . 2. Chú ý :.    1       ; 1;  3  thì Nếu  thỏa mãn  2 2 và m  0;  3    tan   m tan x  m  x  arctan m  k , k . .. 3. Ví dụ minh họa. Ví dụ 3. Giải các phương trình sau .. .     b). 3 tan  2 x    3. c). tan  3 x    1  0. 4 6   3       . d). tan  4 x    tan   2 x   0. e). tan  3 x    tan 2 x  0. f). tan  3 x  300    3 3 4  6   Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. a). tan(3x  )   3. 3. .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(76)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. 4. Câu hỏi trắc nghiệm Mức độ 2. Thông Hiểu.   Câu 12. Số vị trí biểu diễn các nghiệm của phương trình tan  2 x    3  0 trên đường tròn 3  lượng giác là? A. 4 . B. 3 . C. 2 . D. 1 . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 13. (THPT Việt Trì-Phú Thọ 2018) Phương trình tan  x    0 có nghiệm là 3  A. . . 3.  k 2 , k  .. B. . . 2.  k , k  .. C.. . 3.  k , k  .. D. .  3.  k , k . .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 14. (THPT Chuyên Vĩnh Phúc-2018) Phương trình tan x  3 có tập nghiệm là       A.   k 2 , k   . B.  . C.   k , k   . D.   k , k   3  3  6  Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 76. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(77)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản Câu 15. (THPT Kiến An-Hải Phòng 2018)Tìm tất cả các nghiệm của phương trình tan x  m , m   . A. x  arctan m  k hoặc x    arctan m  k ,  k  B. x   arctan m  k ,  k  C. x  arctan m  k 2 ,  k  D. x  arctan m  k ,  k . .. . . .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 16. (THPT Trần Hưng Đạo TP HCM 2018) Giải phương trình A. x  C. x .  3. . 6. k k.  2. . 2. k   .. B. x . k   .. D. x .  3. . 6. 3 tan 2 x  3  0 .  k  k . ..  k  k . .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 17. (THPT Kinh Môn 2 2018) Phương trình A. x .  3.  k 2 .. B. x  .  6.  k .. . . 3 tan x  1  sin 2 x  1  0 có nghiệm là: C. x .  6.  k .. D. x  .  6.  k 2 .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. IV. Phương trình: cot x  m (4) 1. Phương pháp. Điều kiện: x  k  k . ..  . 1   ; ) và m  0;  ; 1;  3  . 2 2 3   Ta đặt cot x  m (4)  cot x  cot   x    k , k  . . Với m    (. 2. Chú ý :.    1       ; 1;  3  thì Nếu  thỏa mãn  2 2 và m  0;  3   cot   m cot x  m  x  arccot m  k , k . 77. Lớp Toán Thầy-Diệp Tuân. .. Tel: 0935.660.880.

<span class='text_page_counter'>(78)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. 3. Ví dụ minh họa. Ví dụ 4. Giải các phương trình sau . 2x      3 a). 2 cot b). 3cot  2 x    3 c). cot  3 x    1  0 4 6 3   Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... 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Câu hỏi trắc nghiệm Mức độ 1. Nhận biết Câu 18. Giải phương trình cot  3x  1   3.. 1 5  k A. x   k  3 18 3 5  k C. x   k  . 18 3. .. 1   B. x    k k  . 3 18 3 1  D. x    k  k   . 3 6. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 78. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(79)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ..................................................................................................................   Câu 19. Với những giá trị nào của x thì giá trị của các hàm số y  tan   x  và y  tan 2 x bằng 4  nhau?. A. x  C. x . . k. 4. . 12. . 2.  k  ..  k  k . . . k  . 12 3    3m  1  ; k, m  . D. x   k k  12 3  2  B. x . .. k. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Mức độ 3. Vận dụng Câu 20. Giải phương trình tan 3x.cot 2 x  1.. . 2. k  .. . . D. Vô nghiệm  k   . C. x  k  k   . 2 Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. A. x  k. B. x  . 4. k. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................     Câu 21. Cho tan  x    1  0 . Tính sin  2 x   . 2 6    1  A. sin  2 x     . 6 2 .  3  . B. sin  2 x    6 2 .  3  . C. sin  2 x     6 2 .  1  D. sin  2 x    . 6 2  Lời giải.. 79. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(80)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 22. Phương trình nào dưới đây có tập nghiệm trùng với tập nghiệm của phương trình tan x  1 ? 2 2 A. sin x  . B. cos x  . C. cot x  1 . D. cot 2 x  1 2 2 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 23. Giải phương trình cos 2 x tan x  0. A. x  k.  2. k  ..    x   k C.  4 2  x  k  .  k  ..   x   k  B. 2   x  k D. x .  2. k  ..  k  k . .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 24. Hỏi trên đoạn 0; 2018  , phương trình A. 6339.. B. 6340.. 3 cot x  3  0 có bao nhiêu nghiệm? C. 2017.. D. 2018.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 80. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(81)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. IV.Mối quan hệ giữa sin x và cos x ; tan x và cot x 1. Phương pháp. Sử dụng công thức phụ chéo để chuyển đổi từ sin x sang cos x và ngược lại. Sử dụng công thức phụ chéo để chuyển đổi từ tan x sang cot x và ngược lại Sử dụng cung đối để đưa dấu trừ vào trong đối với sin x Sử dụng cung bù để đưa dấu trừ vào trong đối với cos x Sử dụng cung hơn kém để đưa dấu trừ vào trong đối với tan x,cot x Hai cung phụ nhau  và. . cos(   )  sin  2 cos( )  cos  sin(   )  sin  tan(   )  tan . .   2. . sin(   )  cos  tan(   )  cot  2 2 Hai cung đối nhau:  và  sin( )   sin  tan( )   tan  Hai cung bù nhau:  và    cos(   )   cos  tan(   )   tan  Hai cung hơn kém nhau  là  và    cot(   )  cot  sin(   )   sin . . cot(   )  tan  2 cot( )   cot  cot(   )   cot  cos(   )   cos . 2. Ví dụ minh họa. Ví dụ 5 . Giải các phương trình sau ..   c). cos  4 x    sin 2 x  0 5  2  9   2  7       d). sin  3 x    cos  x   e). sin  2 x    cos x  0 f). sin  3 x    sin  x  0 3  4  4 3  5             g). tan  3 x    tan 2 x  0 h). tan  3 x    cot x k). cos  3 x    cos x  0 5 3 4    Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. a). sin(2 x  1)  cos(2  x). b). cos(3x  1)  sin(2 x  1). .......................................................................................................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(82)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. 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Ví dụ 6 . Giải các phương trình sau .    4   3x   3 a). sin  3 x    sin  5   5     5   3x   2 c). cos  3 x    sin  3   6 .  4     x   cos   x   3 b). sin   9   18 . d). tan x tan 2 x  1 Lời giải.. 82. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(83)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Câu hỏi trắc nghiệm Mức độ. Nhận biết. x  Câu 25. Gọi X là tập nghiệm của phương trình cos   150   sin x. Mệnh đề nào sau đây là 2  đúng? A. 2900  X . B. 200  X . C. 2200  X . D. 2400  X . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 83. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(84)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. Câu 26.(THPT Chuyên ĐH Vinh-2018) Phương trình tan x  cot x có tất cả các nghiệm là: A. x  C. x . . 4. . 4. k. . 4. k   ..  k 2  k . .. B. x  D. x . . 4. . 4. k. . 2. k   ..  k  k . .. 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VI. Phương trình bậc chẵn. 1. Phương pháp. Sử dụng công thức hạ bậc để đưa về dạng cơ bản. 1  cos 2a 1  cos 2a ⋆ cos 2 a  ⋆ sin 2 a  2 2. ⋆ tan 2 a . 1  cos 2a 1  cos 2a. 2. Ví dụ minh họa. Ví dụ 7. Giải các phương trình sau .  1  a). sin 2  2 x    4 2   3  c). cos 2  2 x    4 4    e). cos 2  3 x    cos 2 x 3 . 2    2  7  x b). sin 2  3 x    sin  3    5      d). sin 2  5 x    cos 2  3 x    0 3 4       f). cos 2  2 x    sin 2  x   4 3  . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 84. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(85)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(86)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. 3. Câu hỏi trắc nghiệm Mức độ. Nhận biết. Câu 27. Trong các phương trình, phương trình nào tương đương với phương trình 2 cos 2 x  1 2 . A. sin x  B. 2sin x  2  0. C. tan x  1. D. tan 2 x  1. 2 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 28. Phương trình nào, có tập nghiệm trùng với tập nghiệm của phương trình tan 2 x  3 ? 1 1 1 . . A. cos x   . B. 4cos 2 x  1. C. cot x  D. cot x   2 3 3 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 29. Giải phương trình 4sin 2 x  3 .    x  3  k 2 ,  k  . A.   x     k 2  3  k  x   C.  3 3  k,  . k  3.    x  3  k 2 , k  B.   x  2  k 2  3 k  x  D.  3  k,  . k  3. .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 30. Trong các phương trình sau, phương trình nào tương đương với 3sin 2 x  cos 2 x ? 3 1 3 . A. sin x  . B. cos x  C. sin 2 x  . D. cot 2 x  3. 2 2 4 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 86. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(87)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 3 có bao nhiêu nghiệm? 4 C. 11. D. 12.. Câu 31. Với x thuộc  0;1 , hỏi phương trình cos 2  6 x   A. 8.. B. 10.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. VII. Tìm tham số m để phương trình có nghiệm. 1. Phương pháp. Phương trình sin x  m , cos x  m có nghiệm khi 1  m  1 .  m  1 Phương trình sin x  m , cos x  m vô nghiệm khi  . m  1 Nếu phương trình bậc hai thì tính  tìm nghiệm đưa về bậc nhất. 2. Ví dụ minh họa. Ví dụ 8. Giải và biện luận phương trình: 2sin( x . . )  2m  1 10 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Ví dụ 9. Giải và biện luận phương trình: m cos 2 x  m  1 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... 87. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(88)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 10. Giải và biện luận các phương trình sau: a). 4sin 2 x  2m  1. . c). tan(2 x  )  m  1 6. . b). (m  1) cos 2 (4 x  )  2m 3. . d). m cot 2 (2 x  )  2m  1 8. 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(89)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... Ví dụ 11. Giải và biện luận các phương trình sau: a). m sin 2 2 x  m  1  0 b). (2m  1) tan 2 3 x  m  2 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Câu hỏi trắc nghiệm. Mức độ 3. Vận dụng Câu 32. Tìm tất các các giá trị thực của tham số m để phương trình sin x  m có nghiệm. A. m  1. B. m  1. C. 1  m  1. D. m  1. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 33. Tìm tất các các giá trị thực của tham số m để phương trình cos x  m  0 vô nghiệm. A. m   ; 1  1;   . B. m 1;   . C. m  1;1. D. m   ; 1 .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 34. Có bao nhiêu giá trị nguyên của tham số m để phương trình cos x  m  1 có nghiệm? A. 1. B. 2. C. 3. D. Vô số Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... 89. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(90)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 35. Gọi S là tập hợp tất cả các giá trị nguyên của tham số m để phương trình   cos  2 x    m  2 có nghiệm. Tính tổng T của các phần tử trong S. 3  A. T  6. B. T  3. C. T  2. D. T  6. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 36. Có tất cả bao nhiêu giá trị nguyên của tham số m để phương trình 3 cos x  m  1  0 có nghiệm? A. 1. B. 2. C. 3. D. Vô số Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 37. Có bao nhiêu giá trị nguyên của tham số m thuộc đoạn  2108; 2018 để phương trình. m cos x  1  0 có nghiệm? A. 2018.. B. 2019.. C. 4036. D. 4038. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 38. Tìm giá trị thực của tham số m để phương trình  m  2  sin 2 x  m  1 nhận x  nghiệm. A. m  2.. 2. .  12. làm. .. 3 1. C. m  4. D. m  1. 32 Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. B. m . .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 39. Tìm tất cả các giá trị của tham số m để phương trình  m  1 sin x  2  m  0 có nghiệm. A. m  1.. 90. 1 B. m  . 2. Lớp Toán Thầy-Diệp Tuân. 1 C. 1  m  . 2. D. m  1.. Tel: 0935.660.880.

<span class='text_page_counter'>(91)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 40. Tìm tất cả các giá trị của tham số m để phương trình  m  2  sin 2 x  m  1 vô nghiệm. 1  A. m   ; 2  . 2 . 1  1  1  B. m   ;    2;   . C. m   ; 2    2;   . D. m   ;   . 2  2  2  Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Dạng VIII. Tìm nghiệm của phương trình nằm trong đoạn  a; b , khoảng  a; b  . 1. Phương pháp. Bước 1: giải phương trình lượng giác cơ bản để tìm họ nghiệm x    k .2 , k  . . a  b  k Bước 2: do x   a; b  a  x  b  a    k .2  b nên giải bất pt 2 2 Bước 3: do k  nên chọn k thỏa mãn rồi thay vào họ nghiệm ban đầu. 2. Ví dụ minh họa. Ví dụ 12. Giải các phương trình sau với điều kiện đã chỉ ra a). 2sin 2 x  1 với. 0  x  2. b). tan 3x   3 với. .  2. x.  2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 91. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(92)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 13. Tìm tổng các nghiệm trong khoảng ( ;  ) của phương. . . a). sin(3x  )  cos(2 x  ) 3 4. . b). sin 2 2 x  cos 2 (3x  ) 8. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(93)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Ví dụ 14. Tìm nghiệm dương nhỏ nhất và nghiệm âm lớn nhất của các phương trình sau: a). sin 2 2 x  cos 2 5 x  1 b). (sin x  cos x) 2  2 cos 2 3 x Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 15. Tìm tổng các nghiệm của phương trình:. . a). 2 cos( x  )  1 trên ( ;  ) 3. . . b). sin(5 x  )  cos(2 x  ) trên [0;  ] 3 3. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 93. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(94)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... . .   Ví dụ 16. Tìm nghiệm nguyên dương của phương trình sin  3 x  9 x 2  16 x  80   0 . 4  Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... 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Ví dụ 17. Tìm nghiệm nguyên dương của phương trình: cos  (3  3  2 x  x 2 )  1 . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. 3. Câu hỏi trắc nghiệm Mức độ. Nhận biết Câu 41. Số nghiệm của phương trình sin  2 x  400   A. 2.. B. 4.. 3 với 1800  x  1800 là 2 C. 6.. D. 7 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 94. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(95)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 3   trên khoảng  ; 2  là? 11 4  A. 1 B. 2. C. 3. D. 4. Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. Câu 42. Số nghiệm của phương trình tan x  tan. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 43. Tổng các nghiệm của phương trình tan 5x  tan x  0 trên nửa khoảng  0;   bằng: A.  .. B.. 3 . 2. C. 2 .. D.. 5 2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 44. Tính tổng T các nghiệm của phương trình sin 2 x  cos x  0 trên 0; 2 . A. T  3 .. B. T . 5 . 2. C. T  2 .. D. T   .. Lời giải.. 95. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(96)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................     Câu 45. Trên khoảng  ; 2  , phương trình cos   2 x   sin x có bao nhiêu nghiệm? 6  2  A. 3 . B. 4 . C. 5 . D. 2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 46. Tổng các nghiệm của phương trình tan  2 x  150   1 trên khoảng  900 ;900  bằng: A. 00.. B. 300.. C. 300.. D. 600.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 96. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(97)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................  3    Câu 47. (THPT Xuân Hòa 2018) Phương trình sin  2 x    sin  x   có tổng các nghiệm 4 4    thuộc khoảng  0;   bằng A.. 7 . 2. B.  .. C.. 3 . 2. D..  4. .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 48.(THPT Hai Bà Trưng-Vĩnh Phúc 2018) Phương trình 2 cos 2 x  1 có số nghiệm trên đoạn  2 ; 2  là A. 2.. B. 4.. C. 6. D. 8. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 5   Câu 49.(THPT Việt Trì-Phú Thọ 2018) Trên đoạn  2 ; , đồ thị hai hàm số y  sin x và 2   y  cos x cắt nhau tại bao nhiêu điểm? A. 2 . B. 5 . C. 4 . D. 3 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 97. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(98)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ...................................................................................................................   Câu 50.(THPT Ninh Giang-Hải Dương 2018) Phương trình 2sin  2 x    3  0 có mấy nghiệm 3  thuộc khoảng  0;3  . A. 6 .. B. 2 .. C. 4 .. D. 8 .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Dạng IX. Phương pháp loại nghiệm khi giải phương trình lượng giác có điều kiện 1. Phương pháp: Phương pháp 1: Biểu diễn các nghiệm và điều kiện lên đưòng tròn lượng giác. Ta loại đi những điểm biểu diễn của nghiệm mà trùng với điểm biểu diễn của điều kiện. Với cách này chúng ta cần ghi nhớ  Điểm biểu diễn cung  và   k 2 , k  trùng nhau 2 k  Để biểu diễn cung   lên đường tròn lượng giác ta cho k nhận n giá trị (thường n chọn k  0,1, 2,..., n  1 ) nên ta có được n điểm phân biệt cách đều nhau trên đường tròn tạo thành một đa giác đều n cạnh nội tiếp đường tròn. Phương pháp 2: Sử dụng phương trình nghiệm nguyên k l Giả sử ta cần đối chiếu hai họ nghiệm   và   , trong đó m, n  đã biết, còn n m k , l  là các chỉ số chạy. k l     ak  bl  c (*) Ta xét phương trình :   n m  Với a, b, c là các số nguyên.  Trong trường hợp này ta quy về giải phương trình nghiệm nguyên ax  by  c (1). Để giải phương trình (1) ta cần chú ý kết quả sau:  Phương trình (1) có nghiệm  d  (a, b) là ước của c b  x  x  t 0  d ,t  .  Nếu phương trình (1) có nghiệm ( x0 ; y0 ) thì (1) có vô số nghiệm  a y  y  0  t Phương pháp 3: Thử trực tiếp  Phương pháp này là ta đi giải phương trình tìm nghiệm rồi thay nghiệm vào điều kiện để kiểm tra. Phương pháp 4: Biểu diễn điều kiện và nghiệm thông qua một hàm số lượng giác:  Giả sử ta có điều kiện là u ( x)  0 ( u ( x)  0, u ( x)  0 ), ta biến đổi phương trình đã cho về phương trình chứa u ( x) và giải phương trình để tìm u ( x) . 2. Ví dụ minh họa. 98. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(99)</span> Trung Tâm Luyện Thi Đại Học Amsterdam Ví dụ 18. Giải các phương trình sau: a). cot 3x  cot x. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản b). cot 4 x.cot 7 x  1. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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.......................................................................................................................................................................................................... ................................................................................................................... sin x cot 5 x 1 cos 9 x Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 19. Giải phương trình sau:. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 99. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(100)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. B. BÀI TẬP RÈN LUYỆN Bài tập 1. Giải các phương trình sau. 1 1 3 a). sin x  600  b). cos 2 x  500  c). tan  3 x  300    3 2 2 3 x  x  x   d). cot   200    e). 1  2cos x  3  cos x   0 f).  cot  1  cot  1  0 3  2  3  2  Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... . . . . .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(101)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. Bài tập 2. Giải các phương trình sau .. . . a). tan  x  300  cos  2 x  1500   0 b). 3tan x  3  2sin x  1  0 d). tan  2 x  600  cos  x  750   0 e).  cot x  1 sin 3x  0.   c). cos 2 x cot  x    0 4  f). tan x tan 2 x  1. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Bài tập 3. Giải các phương trình sau . a). sin  cos x   1.   2  b). 2 cos   sin x  13    3 2    6 . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... 101. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(102)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. C. THỦ THUẬT CASIO. 13    Bài tập 1. Trên đoạn   ; 2  , phương trình cos x  có bao nhiêu nghiệm? 14  2  A. 3. B. 4. C. 5. D. 2. 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Cơ sở lý thuyết: Giá trị hàm số f  x  đổi dấu khi đi qua x  x1 và x  x2 thì phương trình. f  x   0 có một nghiệm trong khoảng  x1 ; x2  . Dựa vào bảng TABLE, ta nhận thấy Ở hàng thứ 4 và hàng thứ 5, f  x  đổi dấu. Suy ra f  x   0 có một nghiệm thuộc  0,392 ; 0  . Ở hàng thứ 5 và hàng thứ 6, f  x  đổi dấu. Suy ra f  x   0 có một nghiệm thuộc  0 ; 0,3926  . Ở hàng thứ 20 và hàng thứ 21, f  x  đổi dấu. Suy ra f  x   0 có một nghiệm thuộ  5,8904 ; 6, 2831 ..    Vậy phương trình đã cho có đúng 3 nghiệm trên đọan   ; 2  .  2 . 102. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(103)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 2. Phương Trình Lượng Giác Cơ Bản.     Bài toán 2. Trên khoảng  ; 2  , phương trình cos   2 x   sin x có bao nhiêu nghiệm? 6  2  A. 3. B. 4. C. 5. D. 2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. TẠO RA SOLVE HỮU HIỆU NHỜ CHỨC NĂNG TABLE.     Bài toán 3. Trên khoảng  ; 2  , tổng T các nghiệm của phương trình cos   2 x   sin x là 6  2  29 37 7 23 . . . . A. T  B. T  C. T   D. T  9 9 9 9 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 103. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(104)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 3. Một Số Phương Trình Lượng Giác Thường Gặp. MỘT SỐ PHƯƠNG TRÌNH LƯỢNG GIÁC THƯỜNG GẶP. §BÀI 3.. A. LÍ THUYẾT I. Phương trình thuần nhất bậc hai đối với sin x, cos x, tan x, cot x . 1. Định nghĩa Phương trình thuần nhất bậc hai đối với sin x, cos x, tan x, cot x là phương trình có cùng một hàm lượng giác (cùng sin x hoặc cùng cos x hoặc cùng tan x hoặc cùng cot x ) với cung góc giống nhau. 2. Phương pháp. Dạng. Đặt ẩn phụ. Điều kiện. a sin 2 x  b sin x  c  0. t  sin x. 1  t  1. a cos 2 x  b cos x  c  0. t  cos x. 1  t  1. a tan 2 x  b tan x  c  0. t  tan x. x. a cot 2 x  b cot x  c  0. t  cot x. x  k. .   k 2. . Chú ý. Nếu đặt t  sin 2 x cos 2 x hoặc t  sin x , cos x thì điều kiện là 0  t  1 . 3. Ví dụ minh họa. Ví dụ 1. Giải các phương trình sau: a). 2sin 2 x  sin x  1  0 c). tan 2 x  2 3 tan x  3  0.. b). 2cos 2 x  3cos x  1  0. d). 3 cot 2 x  (1  3) cot x  1  0.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(105)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 3. Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 2. Giải các phương trình sau: a). 4cos 2 x  4sin x  1  0. c). 2cos2 2 x  5sin 2 x  1  0. e). 3sin 2 x  2cos 4 x  2  0.. b).  sin 2 x  3cos x  3  0. d). 3  4 cos 2 x  sin x(2sin x  1). f). 4sin 4 x  5cos 2 x  4  0.. Nhận xét: Ta áp dụng công thức cos2 x  sin 2 x  1  cos2 x  1  sin 2 x  sin 2 x  1  cos 2 x. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(106)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 3. Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Ví dụ 3. Giải các phương trình sau: a). cos 2 x  3cos x  2  0. x c). 5cos x  2sin  7  0. 2 2 e). cos 2 x  cos x  sin x  2  0.. b). 3cos 2 x  7sin x  2  0. d). sin 2 x  cos 2 x  cos x  2. f). 3cos 2 x  2cos 2 x  3sin x  1.. Nhận xét: áp dụng công thức cos 2 x  2cos 2 x  1  theo cos .  1  2sin 2 x.  theo. sin  ). Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 106. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(107)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 3. Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Ví dụ 4. Giải các phương trình lượng giác sau: a). cos 4 x  12sin 2 x  1  0. b). cos 4 x  2cos 2 x  1  0. x x c). 16sin 2  cos 2 x  15. d). cos 2 x  2 cos x  2sin 2  2 2 x e). cos 2 x  3cos x  4 cos 2  f). 1  cos 4 x  2sin 2 x  0. 2 2 g). 8cos x  cos 4 x  1. h). 6sin 2 3x  cos12 x  4. Nhận xét: Sử dụng kỹ thuật nâng cung, hạ bậc để đưa về cùng góc. (áp dụng công thức cos 2 x  2cos 2 x  1  theo cos .  1  2sin 2 x Hạ bậc cos 2 x .  theo. sin . 1  cos 2 x 1  cos 2 x , sin 2 x  ) 2 2 Lời giải. 107. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(108)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 3. 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(109)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 3. Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 5. Giải các phương trình lượng giác sau: a). 5(1  cos x)  2  sin 4 x  cos 4 x. c). 4(sin 4 x  cos 4 x)  cos 4 x  sin 2 x  0.. b). cos 4 x  sin 4 x  cos 4 x  0. d). 4(sin 6 x  cos 6 x)  4sin 2 x.. 3 Nhận xét: áp dụng công thức cos 4 x  sin 4 x  cos 2 x, cos 6 x  sin 6 x  1  sin 2 2 x 4 1 cos 4 x  sin 4 x  1  sin 2 2 x 2. 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(110)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 3. Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 6. Giải các phương trình lượng giác sau: 3 1 3  3  2 tan 2 x.  3cot 2 x  5.  3cot x  3. a). b). c). 2 2 sin 2 x cos x cos x 4 3 1 2 5  0.    0. d). 9  13cos x  e). 2 tan 2 x  3  f).  tan 2 x  2 1  tan x cos x 2 cos x 2 1  g). 3 sin x  cos x  g). 2sin 2 x  tan 2 x  2. cos x 1 1  1  cot 2 x ,  1  tan 2 x . Nhận xét: áp dụng công thức 2 sin x cos 2 x. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Ví dụ 7. Giải các phương trình lượng giác sau: a). 8sin x cos x  cos 4 x  3  0. cos x  1  sin x. c). 1  sin x Nhận xét: áp dụng công thức sin 2 x  2sin x.cos x .. b). 2sin 2 8 x  6sin 4 x cos 4 x  5. d). 8sin x cos x  cos 4 x  3  0.. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 111. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(112)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 3. Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Ví dụ 8. Giải các phương trình sau . 2     a). cos  2 x    3cos  x    1  0 3  3   2 2 c). 4cos  6 x  2   16cos 1  3x   13.     b). cos 2   x   4 cos   x   4 3  6  d). cos5 x.cos x  cos 4 x.cos 2 x  3cos 2 x  1.       e). 2sin 2  2 x    6sin  x   cos  x    2  0 f). cos 2 3x cos 2 x  cos 2 x  0 3 6 6    Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 112. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(113)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 3. Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(114)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 3. Một Số Phương Trình Lượng Giác Thường Gặp 4. Bài tập vận dụng Bài tập 1. Giải các phương trình sau . a). 4cos 2 x  4sin x  1  0. b). cos 2 x  3cos x  2  0. d). 4sin 4 x  5cos 2 x  4  0.. e). cos 4 x  12sin 2 x  1  0.. c). 3cos 2 x  7sin x  2  0. 1 2 5   0. f).  tan 2 x  2 cos x 2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 2. Giải các phương trình sau . a). tan 2 x . . . 3  1 tan x  3  0 b). cot 2 x  4cot x  3  0. c). tan x  cot x . 3 2. x e). 5cos x  2sin  7  0 f). 23sin x  sin 3x  24 2 Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. d).. 1  cot x  3 sin 2 x. .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Bài tập 3. Giải các phương trình sau . a). sin 3 x  3sin 2 x  2sin x  0. b). cos 4 x  12sin x cos x  5  0. c). cos 2 x  3cos x  4 cos 2. x 2. 5 e). sin 4 x  cos 4 x  1 3 Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. d).. 3  2 3 cot x  6  0 sin 2 x. .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Câu hỏi trắc nghiệm Mức độ 2. Thông hiểu.   Câu 1. Hỏi trên  0;  , phương trình 2sin 2 x  3sin x  1  0 có bao nhiêu nghiệm?  2 A. 1. B. 2. C. 3. D. 4. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 117. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 2. Số vị trí biểu diễn các nghiệm của phương trình 2cos 2 x  5cos x  3  0 trên đường tròn lượng giác là? A. 1. B. 2. C. 3. D. 4. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 3. Cho phương trình cot 2 3x  3cot 3x  2  0. Đặt t  cot x , ta được phương trình nào sau đây? A. t 2  3t  2  0. B. 3t 2  9t  2  0. C. t 2  9t  2  0. D. t 2  6t  2  0. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... . . Câu 4. Số nghiệm của phương trình 4sin 2 2 x  2 1  2 sin 2 x  2  0 trên  0;   là? A. 3.. B. 4.. C. 2.. D. 1. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 118. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(119)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 3. Một Số Phương Trình Lượng Giác Thường Gặp Câu 5. Số nghiệm của phương trình sin 2 2 x  cos 2 x  1  0 trên đoạn   ; 4  là? A. 2.. B. 4.. C. 6.. D. 8.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... x x  3cos  0 trên đoạn 0;8 . 4 4 C. T  16 . D. T  4 .. Câu 6. Tính tổng T tất cả các nghiệm của phương trình 2sin 2 A. T  0.. B. T  8 .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 7. Số nghiệm của phương trình A. 1.. B. 2.. 1  sin 2 x. . . 3  1 cot x . . C. 3.. . 3  1  0 trên  0;   là? D. 4.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 119. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 8. Tính tổng T tất cả các nghiệm của phương trình 2 cos 2 x  2 cos x  2  0 trên đoạn 0;3  . A. T . 17 . 4. B. T  2 .. C. T  4 .. D. T  6 .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 9. Số vị trí biểu diễn các nghiệm của phương trình cos 2 x  3sin x  4  0 trên đường tròn lượng giác là? A. 1. B. 2. C. 3. D. 4. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... x x Câu 10. Cho phương trình cos x  cos  1  0 . Nếu đặt t  cos , ta được phương trình nào sau 2 2 đây? A. 2t 2  t  0. B. 2t 2  t  1  0. C. 2t 2  t  1  0. D. 2t 2  t  0. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 120. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Mức độ 3. Vận dụng.     5 Câu 11. Số nghiệm của phương trình cos 2  x    4 cos   x   thuộc  0; 2  là? 3  6  2 A. 1. B. 2. C. 3. D. 4. 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Câu 12. (Chuyên Biên Hòa Hà Nam 2018) Số nghiệm của phương trình 2sin 2 2 x  cos 2 x  1  0 trong  0; 2018 là A. 1008 .. B. 2018 .. C. 2017 . D. 1009 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 13. (THPT Can Lộc 2018) 9  15    Số nghiệm của phương trình sin  2 x    3cos  x    1  2sin x với x   0;2 là: 2  2    A. 6 . B. 5 . C. 3 . D. 4. 121. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Câu 14.(Sở GD&ĐT Hà Tĩnh 2018) Tổng các nghiệm của phương trình 2 cos 2 x  3 sin 2 x  3 trên  5   0;  là:  2  7 7 7 A. . B. . C. . D. 2 6 3 2 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Mức độ 4. Vận dụng cao Câu 15. Tìm tất cả các giá trị thực của tham số m để phương trình tan x  m cot x  8 có nghiệm. A. m  16. B. m  16. C. m  16. D. m  16. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 122. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 16. Tìm tất cả giá trị thực của tham số m để phương trình cos 2 x   2m  1 cos x  m  1  0 có   3  nghiệm trên khoảng  ;  . 2 2 . A. 1  m  0 .. B. 1  m  0 .. C. 1  m  0 .. D. 1  m . 1 2. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 17. Biết rằng khi m  m0 thì phương trình 2sin 2 x   5m  1 sin x  2m2  2m  0 có đúng 5.    nghiệm phân biệt thuộc khoảng   ;3  . Mệnh đề nào sau đây là đúng?  2  1 3 7   3 2 A. m  3. B. m  . C. m0   ;  . D. m0    ;   . 2  5 10   5 5 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 123. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 18. Tìm tất cả các giá trị thực của tham số m để phương trình.    2cos 2 3x   3  2m  cos3x  m  2  0 có đúng 3 nghiệm thuộc khoảng   ;  .  6 3 A. 1  m  1. B. 1  m  2. C. 1  m  2. D. 1  m  2. 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II. Phương trình bậc nhất đối với sin x, cos x . 1. Định nghĩa. Phương trình bậc nhất đối với sin x, cos x là phương trình có dạng: a sin x  b cos x  c (1) ; với. a, b, c . và a 2  b2  0 .. 2. Phương pháp. Chia hai vế 1 cho và đặt cos  . a 2  b 2 ta được. a. ;sin  . a a 2  b2. sin x . b a 2  b2. cos x . c a 2  b2. (). b. . a  b2 c c  ()  sin x.cos   cos x.sin    sin( x   )  (2). 2 2 2 a b a  b2. 124. a b 2. 2. Lớp Toán Thầy-Diệp Tuân. 2. Tel: 0935.660.880.

<span class='text_page_counter'>(125)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 3. Một Số Phương Trình Lượng Giác Thường Gặp 3. Chú ý: có nghiệm  (2) có nghiệm  a 2  b2  c 2 .. 1  3  sin x  3 cos x  2  sin x  cos x   2sin( x  ) 2 3 2   3  1  3 sin x  cos x  2  sin x  cos x   2sin( x  ) 2 6  2  1   1  sin x  cos x  2  sin x  cos x   2 sin( x  ) . 4 2  2  4. Công thức bỗ trợ. Công thức cộng cos  a  b   cos a cos b sin a sin b. sin 2a  2sin a cos a. sin  a  b   sin a cos b  sin b cos a. Công thức nhân đôi cos 2a  2 cos 2 a  1  theo cos .  1  2sin 2 a.  theo sin .  cos 2 a  sin 2 a.  theo tong . 5. Ví dụ minh họa. Ví dụ 9. Giải các phương trình sau . a). cos x  3 sin x  2 b). 3sin x  4cos x  5 d). sin 2 x  3 cos 2 x  1 1. tan( a  b) . e). sin 3 x  3 cos 3 x  2sin 2 x. tan a  tan b 1 tan a. tan b. cos3a  4cos3 a  3cos a sin 3a  3sin a  4sin 3 a. c). 3 sin x  cos x  2 2 f). 2sin x  3 sin 2 x  3. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Ví dụ 10. Giải các phương trình sau . a). sin x  cos x  2 2 sin x.cos x   c). sin   2 x   3 sin   2 x   2 2  3  3 cos 2 x  cos x e). 2sin x.   b). 3 sin 7 x  cos 7 x  2sin  5 x   6    d). cos x  3 sin x  2 cos  2 x    0 3  3 f). 3sin 3 x  3 cos 9 x  1  4sin 3 x. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... 126. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Ví dụ 11. Giải các phương trình sau . a). sin 8 x  cos 6 x  3  sin 6 x  cos8 x . b). sin x  sin 2 x  3  cos x  cos 2 x . 2. x x  c).  sin  cos   3 cos x  2 2 2    e). 2sin  2 x    4sin x  1  0 6 . d). f).. 3 cos 5 x  2sin 3 x cos 2 x  sin x  0. 1  2sin x  cos x  1  2sin x 1  sin x . 3 1. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 128. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Ví dụ 12. Giải các phương trình sau ..   1   b). 2sin  x    sin  2 x    3 6 2   3 1  c). 2cos2 x  2 3 sin x cos x  1  3(sin x  3 cos x) d). 8sin x  cos x sin x   3 3 2 2 e). sin x  3 cos x  sin x cos x  3 sin x cos x f). 2 cos  2 x    4sin x cos x  1  0 6  a). sin 3 x  3 cos 3 x  2sin 2 x. 129. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Câu hỏi trắc nghiệm Mức độ 3. Vận dụng. Câu 19. Gọi S là tập nghiệm của phương trình cos 2 x  sin 2 x  1 . Khẳng định nào sau đây là đúng?   3 5  S.  S. A.  S . B.  S . C. D. 4 2 4 4 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 20. Số nghiệm của phương trình sin 2 x  3 cos 2 x  3 trên khoảng  0;  là?  2 A. 1. B. 2. C. 3. D. 4. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... 131. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 21. Tính tổng T các nghiệm của phương trình cos 2 x  sin 2 x  2  sin 2 x trên  0; 2  . A. T . 7 . 8. B. T . 21 . 8. C. T . 11 . 4. D. T . 3 . 4. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 22. Tìm nghiệm dương nhỏ nhất x0 của 3sin 3 x  3 cos 9 x  1  4sin 3 3 x. A. x0 .  2. .. B. x0 .  18. .. C. x0 .  24. .. D. x0 .  54. .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 132. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 23. Số nghiệm của phương trình sin 5 x  3 cos 5 x  2sin 7 x trên khoảng  0;  là?  2 A. 2. B. 1. C. 3. D. 4. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Mức độ 4. Vận dụng cao.     3 cos  x    sin  x    2sin 2 x. 2 2   5 7    x  6  k 2  x  6  k 2 A.  B.  , k . , k .  x    k 2  x     k 2   18 3 18 3 5  2    x  6  k 2  x  18  k 3 C.  D.  , k . , k .  x  7  k 2  x     k 2   6 18 3 Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. Câu 24. Giải phương trình. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 133. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 25. Gọi x0 là nghiệm âm lớn nhất của sin 9 x  3 cos 7 x  sin 7 x  3 cos 9 x . Mệnh đề nào sau đây là đúng?             A. x0    ;0  . B. x0    ;   . C. x0    ;   . D. x0    ;   .  12   6 12   3 6  2 3 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 26. Biến đổi phương trình cos3x  sin x  3  cos x  sin 3x  về dạng sin  ax  b   sin  cx  d     với b , d thuộc khoảng   ;  . Tính b  d .  2 2. A. b  d . . 12. .. B. b  d . . 4. .. . C. b  d   . 3. D. b  d .  2. .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 134. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 27. Giải phương trình. A. x . . cos x  3 sin x  0. 1 sin x  2.  k , k  .. 6 7  k 2 , k  . C. x  6. .  k 2 , k  . 6 7  k , k  . D. x  6 B. x . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 2sin 2 x  cos 2 x có tất cả bao nhiêu giá trị nguyên? sin 2 x  cos 2 x  3 A. 1. B. 2. C. 3. D. 4. Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. Câu 28. Hàm số y . .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 135. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(136)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 3. Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 29. Gọi x0 là nghiệm dương nhỏ nhất của cos 2 x  3 sin 2 x  3 sin x  cos x  2. Mệnh đề nào sau đây là đúng?             A. x0   0;  . B. x0   ;  . C. x0   ;  . D. . x0   ;  .  6 3  3 2  12  12 6  Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 30. Có bao nhiêu giá trị nguyên của tham số m thuộc đoạn  10;10 để phương trình. A. 21..     sin  x    3 cos  x    2m vô nghiệm. 3 3   C. 18. D. 9.. B. 20.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 31. Tìm tất cả các giá trị thực của tham số m để phương trình cos x  sin x  2  m 2  1 vô nghiệm. A. m   ; 1  1;   .. B. m  1;1.. C. m   ;  . D. m   ;0    0;   . Lời giải.. 136. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 32. Có bao nhiêu giá trị nguyên của tham số m thuộc đoạn  10;10 để phương trình.  m  1 sin x  m cos x  1  m có nghiệm. A. 21.. B. 20.. C. 18.. D. 11.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 33. Có bao nhiêu giá trị nguyên của tham số m thuộc đoạn  2018; 2018 để phương trình.  m  1 sin 2 x  sin 2 x  cos 2 x  0 A. 4037.. B. 4036.. C. 2019.. có nghiệm.. D. 2020.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 34. (THPT Chuyên Phan Bội Châu 2018) Có bao nhiêu giá trị nguyên dương của tham số m để phương trình cos 2 x  m sin x  m  0 có nghiệm? A. 0 .. B. 1 .. C. 2 .. D. Vô số. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 137. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 35. (THPT Can Lộc Hà Tĩnh 2018) Tổng tất cả các giá trị nguyên của m để phương trình 4sin x   m  4  cos x  2m  5  0 có nghiệm là: A. 5 .. B. 6 .. C. 10 .. D. 3 .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 36. (Toán Học Tuổi trẻ 2018) Với giá trị lớn nhất của a bằng bao nhiêu để phương trình a sin 2 x  2sin 2 x  3a cos 2 x  2 có nghiệm? 11 8 A. 2 . B. . C. 4 . D. 3 3 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 138. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(139)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 3. Một Số Phương Trình Lượng Giác Thường Gặp. .......................................................................................................................................................................................................... .................................................................................................................. Câu 37.(THPT Đặng Thúc Hứa Nghệ An 2018) Gọi S là tập hợp các nghiệm thuộc khoảng 2. x x  0;100  của phương trình  sin  cos   3 cos x  3 . Tổng các phần tử của S là 2 2  7525 7550 7400 7375 A. . B. . C. . D. 3 3 3 3 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(140)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác III. Phương trình đẳng cấp bậc hai đối với sin x,cos x. 1. Định nghĩa : Phương trình đẳng cấp bậc hai đối với sin x,cos x là phương trình có dạng. a sin 2 x  b sin x cos x  c cos 2 x  d 1 , a, b, c, d  2. Phương pháp: Xét 2 trường hợp : Trường hợp 1: Xét cos x  0  sin x  1 .  Thay vào (1) xem thoả hay không thoả rồi kết luận. Trường hợp 2: Xét cos x  0.  Chia hai vế của (1) cho cos 2 x , rồi đưa về phương trình bậc hai theo tan x có dạng 1   a  d  tan 2 x  b tan x  c  d  0 rồi giải bình thường. 3. Nhận xét: Ta có thể mở rộng ra bậc k . Là phương trình có dạng f (sin x, cos x)  0 trong đó luỹ thừa của sin x và cos x cùng chẵn hoặc cùng lẻ. Cách giải: Chia hai vế phương trình cho cos k x  0 ( k là số mũ cao nhất) ta được phương trình ẩn là tan x . 4. Công thức bỗ trợ.. 1 cos 2  sin 3a  3sin a  4sin 3 a. 1  tan 2  . sin 2   cos 2   1 sin 2a  2sin a cos a. 1 sin 2  cos3a  4cos3 a  3cos a. 1  cot 2  . 5. Ví dụ minh họa. Ví dụ 13. Giải các phương trình sau . a). 2sin 2 x  3 3 sin x cos x  cos 2 x  4. b). 3sin 2 2 x  sin 2 x cos 2 x  4cos 2 2 x  2 x x c). 2sin 2 x  3  3 sin x cos x  3  1 cos 2 x  1 d). 3sin 2  4sin x  8 3  9 cos 2  0 2 2 2 2 2 2 e). 3 sin x  1  3 sin x cos x  cos x  1  3  0 f). 9sin x  30sin x cos x  25cos x  25. . . . . . . . . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 140. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(141)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Ví dụ 14. Giải các phương trình sau . 1 a). sin 2 x  sin 2 x  2 cos 2 x  2 3 c). 2sin x  cos x e). 6sin x  2cos3 x  5sin 2 x cos x. b). sin 2 x  2sin 2 x  2cos 2 x d). 3sin 3 x  2sin 2 x cos x  sin x cos 2 x f). sin x  4sin 3 x  cos x  0. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 141. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(142)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(143)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 15. Giải các phương trình sau . a). 4  sin 3 x  cos3 x   cos x  3sin x. b). 3cos 4 x  4sin 2 x cos 2 x  sin 4 x  0.   d). 2 2 cos3  x    3cos x  sin x  0 4    e). sin 3 x  4sin 2 x cos x  5sin x cos 2 x  2cos3 x  0 f). 8cos3  x    cos 3 x 3  Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. c). tan x.sin 2 x  2sin 2 x  3  cos 2 x  sin x.cos x . .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(144)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(145)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. 6. Câu hỏi trắc nghiệm Câu 38. Giải phương trình sin 2 x  A. x . .  k 2  k . 3    x  3  k 2 C.   x    k 2  4. . .. k  .. Mức độ 2. Thông hiểu. . 3  1 sin x cos x  3 cos 2 x  0.. .  k  k   . 4    x  3  k D.   k  .  x    k  4 B. x . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 39. Gọi S là tập nghiệm của phương trình 2sin 2 x  3 3 sin x cos x  cos 2 x  2 . Khẳng định nào sau đây là đúng?        5    5  A.  ;    S . B.  ;   S . C.  ;   S . D.  ;   S . 3  6 2  4 12  2 6  Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 40. Trong các phương trình sau, phương trình nào tương đương với phương trình sin 2 x  3  1 sin x cos x  3 cos 2 x  3 .. . . A. sin x  0 ..   B. sin  x    1 . 2 .  3 1  C.  cos x  1  tan x    0 . 1  3  . D. tan x  2  3  cos 2 x  1  0. 145. Lớp Toán Thầy-Diệp Tuân. . . Tel: 0935.660.880.

<span class='text_page_counter'>(146)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 41. Phương trình nào dưới đây có tập nghiệm trùng với tập nghiệm của phương trình sin 2 x  3 sin x cos x  1 ?      A. cos x  cot 2 x  3  0 . B. sin  x   .  tan  x    2  3   0 . 2   4  .     C. cos 2  x    1 . tan x  3  0 . 2   . . . . . D.  sin x  1 cot x  3  0 .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 42. Cho phương trình cos 2 x  3sin x cos x  1  0 . Mệnh đề nào sau đây là sai? A. x  k không là nghiệm của phương trình. B. Nếu chia hai vế của phương trình cho cos 2 x thì ta được phương trình tan 2 x  3tan x  2  0 . C. Nếu chia 2 vế của phương trình cho sin 2 x thì ta được phương trình 2cot 2 x  3cot x  1  0 . D. Phương trình đã cho tương đương với cos 2 x  3sin 2 x  3  0 . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 146. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(147)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 43. Số vị trí biểu diễn các nghiệm phương trình sin 2 x  4sin x cos x  4cos 2 x  5 trên đường tròn lượng giác là? A. 4 . B. 3 . C. 2 . D. 1 . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 44. Số nghiệm của phương trình cos 2 x  3sin x cos x  2sin 2 x  0 trên  2 ; 2  ? A. 2 .. B. 4 .. C. 6 . D. 8 . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 45. Nghiệm dương nhỏ nhất của phương trình 4sin 2 x  3 3 sin 2 x  2 cos 2 x  4 là: A.. . 12. .. B.. . 6. .. C.. . 4. .. D..  3. .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 147. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(148)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 46. Cho phương trình. . . 2  1 sin 2 x  sin 2 x . . . 2  1 cos 2 x  2  0 . Trong các mệnh đề sau,. mệnh đề nào sai? 7 A. x  là một nghiệm của phương trình. 8 B. Nếu chia hai vế của phương trình cho cos 2 x thì ta được phương trình tan 2 x  2 tan x  1  0 . C. Nếu chia hai vế của phương trình cho sin 2 x thì ta được phương trình cot 2 x  2cot x  1  0 . D. Phương trình đã cho tương đương với cos 2 x  sin 2 x  1 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... . . . . Câu 47. Giải phương trình 2sin 2 x  1  3 sin x cos x  1  3 cos 2 x  1. A. .  . 6. B. .  . 4. C. . 2 . 3. D. .  . 12. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 48. Có bao nhiêu giá trị nguyên của tham số m thuộc đoạn  10;10 để phương trình. 11sin 2 x   m  2  sin 2 x  3cos 2 x  2 có nghiệm? A. 16.. B. 21.. C. 15.. D. 6.. Lời giải.. 148. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(149)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 49. Có bao nhiêu giá trị nguyên của tham số m thuộc để phương trình sin 2 x  2  m  1 sin x cos x   m  1 cos 2 x  m có nghiệm? A. 2.. B. 1.. C. 0.. D. Vô số. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 50. Tìm điều kiện để phương trình a sin 2 x  a sin x cos x  b cos 2 x  0 với a  0 có nghiệm. 4b 4b  1. 1 A. a  4b . B. a  4b . C. D. a a Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 51. Tìm tất cả các giá trị của tham số m để phương trình 2sin 2 x  m sin 2 x  2m vô nghiệm. 4 4 4 4 A. 0  m  . B. m  0 , m  . C. 0  m  . D. m   , m  0 3 3 3 3 Lời giải.. 149. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 52. Có tất cả bao nhiêu giá trị nguyên của tham số m thuộc đoạn  3;3 để phương trình. m. 2.  2  cos 2 x  2m sin 2 x  1  0 có nghiệm.. A. 3 .. B. 7 .. C. 6 .. D. 4. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. IV. Phương trình đối xứng (phản đối xứng) đối với sin x,cos x. 1. Định nghĩa Phương trình đối xứng đối với sin x,cos x là phương trình có dạng. a(sin x  cos x)  b sin x cos x  c  0.  4 , a, b, c . 2. Phương pháp: Để giải phương trình trên ta sử dụng phép đặt ẩn phụ: t 2 1  sin x cos x    Đặt t  sin x  cos x  2 sin  x     2 4    t   2; 2    Thay vào (4) ta được phương trình bậc hai theo t.. 150. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(151)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. 3. Nhận xét: Ngoài ra chúng ta còn gặp phương trình phản đối xứng có dạng. a(sin x  cos x)  b sin x cos x  c  0.  4'. t    2; 2       Để giải phương trình này ta cũng đặt t  sin x  cos x  2 sin  x     2 4  sin x cos x  1  t   2 Thay vào (4’) ta có được phương trình bậc hai theo t.. 4. Ví dụ minh họa. Ví dụ 16. Giải các phương trình sau . a). 2sin 2 x  3 3  sin x  cos x   5  0. b). 2(sin x  cos x)  6sin x cos x  2  0.. c). 2 2  sin x  cos x   2sin 2 x  1. d). sin x  cos x  4sin x cos x 1  0.. e). sin x cos x  2(sin x  cos x)  1  0.. f). 2sin 2 x  3 3(sin x  cos x)  5  0.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(153)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 17. Giải các phương trình sau . a). sin x  cos x  2 6 sin x cos x.. b). 2 2(sin x  cos x)  3  sin 2 x. 1 1 c). (1  2)(1  sin x  cos x)  sin 2 x d).   2 2. sin x cos x 1 e). (1  2)(sin x  cos x)  2sin x cos x  1  2  0. f). sin x  2sin 2 x   cos x. 2 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(154)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Ví dụ 18. Giải các phương trình sau .   a). sin 2 x  2 sin  x    1. 4  cos 2 x 1 c). cot x  1   sin 2 x  sin 2 x 1  tan x 2. b). 2sin 2 x  3 6 sin x  cos x  8  0. d).. 1 1     2 2 cos  x   cos x sin x 4 . 1. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 154. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Câu hỏi trắc nghiệm Mức độ 3. Vận dụng Câu 53.(SGD Bà Rịa Vũng Tàu 2018) Cho x0 là nghiệm của phương trình sin x cos x  2  sin x  cos x   2 thì giá trị của P  3  sin 2 x0 là A. P  3 .. B. P  3 . 2 . 2. C. P  0 .. D. P  2 .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 155. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 54.(THPT Chuyên Bắc Ninh 2018) Giải phương trình sin 3x  4sin x cos 2 x  0. k 2 k    x  k 2  x  k x  3 x  2  A.  B.  C. D.     x     k 2   x    k  x     k x    k 3 6     3 4 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 55.(SGD Bà Rịa Vũng Tàu 2018).   Cho x0 là nghiệm của phương trình sin x cos x  2  sin x  cos x   2 thì giá trị của P  sin  x0   4  1 2 2 A. P  . B. P  1 . C. P  . D. P   2 2 2 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 156. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(157)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... .................................................................................................................. Câu 56.(THPT Phan Châu Trinh-2018) Tổng các nghiệm của phương trình sin x cos x  sin x  cos x  1 trên khoảng  0; 2  là A. 2 .. B. 4 .. C. 3 .. D.  .. 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Câu 57.(THPT Chuyên Lam Sơn 2018) Có bao nhiêu giá trị nguyên của tham số m để phương m trình: 1  2 cos x  1  2sin x  có nghiệm thực. 2 A. 3 . B. 5 . C. 4 . D. 2 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Câu 58.(THPT Phan Chu Trinh 2018) Tổng các nghiệm của phương trình sin x cos x  sin x  cos x  1 trên khoảng  0;2  là: A. 2 . B. 4 . C. 3 . D. . 157. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(158)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. V. Phương trình biến đổi tổng thành tích, tích thành tổng, hạ bậc (bậc chẵn) 1. Phương pháp . Khi gặp phương trình có chứa tổng (hiệu) của hai hay nhiều hàm lượng giác ta sử dụng công thức biến đổi tổng thành tích để xuất hiện thừa số chung đưa về phương trình tích:  f  x  0  g  x  0 f  x  .g  x  ....h  x   0   ..............   h  x   0. 2. Công thức bổ trợ. ab a b ab a b .cos cos a  cos b  2sin .sin 2 2 2 2 ab a b ab a b sin a  sin b  2sin .cos sin a - sin b  2 cos .sin 2 2 2 2 Khi gặp phương trình có chứa tích của hai hàm lượng giác ta sử dụng công thức biến đổi tích thành tổng để xuất hiện dạng phương trình lượng giác cơ bản. cos a  cos b  2 cos. 1 1 cos a.cos b  [cos(a  b)  cos(a  b)] sin a.sin b  [cos(a  b)  cos(a  b)] 2 2 1 1 sin a.cos b  [sin(a  b)  sin(a  b)] cos a sin b.  [sin(a  b)  sin(a  b)] 2 2 Khi gặp phương trình có chứa hàm lượng giác bậc chẵn ta sử dụng công thức hạ bậc để đưa về tổng của hay hay nhiều hàm lường giác rồi tiếp tục biến đổi tổng tích thành để xuất hiện thừa số chung. 1  cos 2a  1  cos 2a  2 cos 2 a 2 1  cos 2a tan 2 a  1  cos 2a 3.Ví dụ minh họa : cos 2 a . Ví dụ 19. Giải các phương trình sau . a). sin 3x  cos 2 x  sin x  0 c). sin x  sin 2 x  sin 3x  cos x  cos 2 x  cos3x e). sin 3x  cos 3x  sin x  cos x  2 cos 2 x. 1  cos 2a  1  cos 2a  2sin 2 a 2 2 1  sin 2a   sin x  cos x  sin 2 a . b). cos3x  cos 2 x  cos x 1  0 d). 2sin 2 2 x  sin 7 x  1  sin x f). cos3x  2sin 2 x  cos x  sin x  1  0.  . Lời giải.. 158. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(159)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(160)</span> Trung Tâm Luyện Thi Đại Học Amsterdam Ví dụ 20. Giải các phương trình sau . a). sin 4 x sin 7 x  cos3x cos 6 x c). 2cos5x.cos3x  sin x  cos8 x. Chương I-Bài 3. Một Số Phương Trình Lượng Giác b). cos 2 x.cos x  cos x  sin 2 x.sin x d). sin 2 x.cos3x  sin 5x.cos 6 x. 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Ví dụ 21. Giải các phương trình sau . a). sin 2 4 x  sin 2 3x  sin 2 2 x  sin 2 x c). sin 2 3x  cos 2 4 x  sin 2 5 x  cos 2 6 x     4  sin x e). cos 2   x   cos 2   x   2 3  3 . b). sin 2 x  sin 2 3x  cos 2 2 x  cos 2 4 x d). cos 2 3x cos 2 x  cos 2 x  0. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 160. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(161)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Câu hỏi trắc nghiệm Mức độ 3. Vận dụng Câu 59. Giải phương trình sin x cos x  2  sin x  cos x   2 ..   x   k  , k . A. 2   x  k   x    k 2 , k . C.  2   x  k 2.   x   k 2  , k . B. 2   x  k 2   x    k , k . D.  2   x  k Lời giải.. 161. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(162)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 60. Cho phương trình 3 2  sin x  cos x   2sin 2 x  4  0 . Đặt t  sin x  cos x , ta được phương trình nào dưới đây? A. 2t 2  3 2 t  2  0.. B. 4t 2  3 2 t  4  0.. C. 2t 2  3 2 t  2  0.. D. 4t 2  3 2 t  4  0.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 61. Cho phương trình 5sin 2 x  sin x  cos x  6  0 . Trong các phương trình sau, phương trình nào tương đương với phương trình đã cho?  2  3   . . A. sin  x    B. cos  x    C. tan x  1. D. 1  tan 2 x  0. 4 2 4 2   Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 162. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(163)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... 1 Câu 62. Nghiệm âm lớn nhất của phương trình sin x  cos x  1  sin 2 x là: 2  3 A.  . B.   . C.  . D.  2 . 2 2 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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.......................................................................................................................................................................................................... ...................................................................................................................   Câu 63. Cho x thỏa mãn phương trình sin 2 x  sin x  cos x  1. Tính sin  x   . 4      A. sin  x    0 hoặc sin  x    1 . 4 4  .  2  C. sin  x     . 4 2 .  2    B. sin  x    0 hoặc sin  x    . 4 4 2    2    D. sin  x    0 hoặc sin  x     4 4 2  . .. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 163. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(164)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác.   Câu 64. Từ phương trình 5sin 2 x  16  sin x  cos x   16  0 , tìm được sin  x   có giá trị bằng: 4  2 2 2 . . . A. B.  C. 1. D.  2 2 2 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 65. Cho x thỏa mãn 6  sin x  cos x   sin x cos x  6  0 . Tính cos  x   . 4      A. cos  x    1. B. cos  x    1. 4 4    1  1   . . C. cos  x    D. cos  x     4 4 2 2   Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 164. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(165)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... . . Câu 66. Từ phương trình 1  3  cos x  sin x   2sin x cos x  3  1  0 , nếu ta đặt t  cos x  sin x. thì giá trị của t nhận được là: A. t  1 hoặc t  2 . B. t  1 hoặc t  3 . C. t  1 . D. t  3 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... . . Câu 67. Nếu 1  5  sin x  cos x   sin 2 x  1  5  0 thì sin x bằng bao nhiêu? 2 . 2 C. sin x  1 hoặc sin x  0 .. A. sin x . 2 2 hoặc sin x   . 2 2 D. sin x  0 hoặc sin x  1.. B. sin x . Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   Câu 68. Nếu 1  sin x 1  cos x   2 thì cos  x   bằng bao nhiêu? 4  2 2 . . A. 1. B. 1. C. D.  2 2 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 165. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(166)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 69. Cho x thỏa mãn 2sin 2 x  3 6 sin x  cos x  8  0 . Tính sin 2 x.. 1 A. sin 2 x   . 2. 1 2 2 . . C. sin 2 x  . D. sin 2 x  2 2 2 Lời giải. .......................................................................................................................................................................................................... .................................................................................................................. B. sin 2 x  . .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 70. Hỏi trên đoạn 0; 2018  , phương trình sin x  cos x  4sin 2 x  1 có bao nhiêu nghiệm? A. 4037.. B. 4036.. C. 2018.. D. 2019.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 166. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(167)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 71. Từ phương trình A. 1.. 2  sin x  cos x   tan x  cot x , ta tìm được cos x có giá trị bằng: B. . 2 . 2. C.. 2 . 2. D.  1.. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 3   Câu 72. Từ phương trình 1  sin 3 x  cos3 x  sin 2 x , ta tìm được cos  x   có giá trị bằng: 4 2  2 2 2 . . . A. 1. B.  C. D.  2 2 2 Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 167. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(168)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 73. Có bao nhiêu giá trị nguyên của tham số m để phương trình sin x cos x  sin x  cos x  m  0 có nghiệm? A. 1. B. 2. C. 3. D. 4. 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(169)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. III. Phương trình Tích 1. Phương pháp: Khi gặp một bài toán phương trình lượng giác mà không có các dạng trên ta phải sử dụng các phương pháp phân tích đa thức thành nhân tử như: Đặt thừa số chung. Hằng đẳng thức Nhóm hạng tử Thêm bớt, tách hạng tử. Để đưa phương trình về dạng phương trình tích. 2. Công thức bổ trợ. Ta thường chú ý các công thức sau 1  cos 2 x  2cos 2 x 1  cos 2 x  2sin 2 x 3. Ví dụ minh họa. Ví dụ 22. Giải các phương trình sau . a). sin 5 x  2cos 2 x  1 c). cos 2 x.cos x  cos x  sin 2 x.sin x  e). 1  sin x  1  sin x  .sin 2 x  cos 2 x g). cos3x  2sin 2 x  cos x  sin x  1  0. b). sin 3x  cos 2 x  sin x  0 d). 2cos5 x.cos3x  sin x  cos8 x .    . f). 2sin 3 x  cos 2 x  cos x  0.  . h). (1  2sin x) 2 cos x  1  sin x  cos x. 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(170)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(171)</span> Trung Tâm Luyện Thi Đại Học Amsterdam Ví dụ 23. Giải các phương trình sau . a). 1  sin x   sin x  cos x  cos x c). cos 3x  4cos 2 x  3cos x  4  0, x   0;14. Chương I-Bài 3. Một Số Phương Trình Lượng Giác b). 1  sin x  sin 2 x  cos x  cos 2 x  0 d).  2cos x  1 2sin x  cos x   sin 2 x  sin x 1. e). 1  sin  cos x  sin 2 x  cos 2 x  0 f). sin 2 x  cos 2 x  3sin x  cos x  2  0 g). sin x  sin 2 x  sin 3x  cos x  cos 2 x  cos3x h). sin 2 x  2 cos x  3 sin x  3 1. 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.......................................................................................................................................................................................................... ................................................................................................................... 171. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(172)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Ví dụ 24. Giải các phương trình sau . 3x a). cos 2 x  cos x  2sin 2 2 1  cos 2 x sin 2 x  c). cos x 1  cos 2 x 3 3 e). cos x  sin x  2sin 2 x  1 (1). b). sin x  cos x . cos 2 x 1  sin 2 x. d). cos 2 x  1  2cos x  sin x  cos x   0. f). 4sin 3 x  4sin 2 x  3sin 2 x  6cos x  0 (1) Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 172. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(173)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Ví dụ 25. Giải các phương trình sau . a). 1  sin 2 x  .cos x  1  cos 2 x  .sin x  1  sin 2 x. b).  sin 2 x  cos 2 x  .cos x  2cos 2 x  sin x  0. c). sin 3 x  3 cos3 x  sin x.cos 2 x  3 sin 2 x.cos x d). sin 2 x  cos 2 x  3sin x  cos x  1  0 1 Lời giải.. 173. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(174)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(175)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Ví dụ 26. Giải các phương trình sau . a). tan x  tan 2 x  sin 3x.cos x c). 2sin 3 x  cos 2 x  sin x e).  2sin 2 x  1 tan 2 2 x  3  2 cos 2 x  1  0 (1) g). 2sin 2 2 x  sin 7 x  1  sin x. b). cos 2 x  sin 2 x  sin 3x  cos 4 x 1 d). sin x.sin 2 x.sin 3x  sin 4 x 4 2 f). 1  sin x  cos x  1  cos 2 x  sin x  1  sin 2 x h). 4  sin 4 x  cos 4 x   cos 4 x  sin 2 x  0. 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.......................................................................................................................................................................................................... ................................................................................................................... 175. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(176)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. 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Ví dụ 27. Giải các phương trình sau . a). 3 sin 2 x  cos 2 x  2 cos x  1. . . b). 3 cos 2 x  2 cos x  sin x  1  0. c). 2 cos x  3 sin x cos x  cos x  3 sin x  1 d). sin 3x  cos 3x  sin x  cos x  2 cos 2 x e). 3 sin 2 x  cos 2 x  1  3 sin x  3cos x g). 2sin 2 x  sin 2 x  sin x  cos x  1  0. 176. Lớp Toán Thầy-Diệp Tuân. f). sin x  cos x sin 2 x  3 cos 3x  2  cos 4 x  sin 3 x  h)..   2 sin  2 x    2sin x  1 4 . Tel: 0935.660.880.

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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(178)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Ví dụ 28. Giải các phương trình sau .   2   a). sin  2 x    sin  x    4 4 2   c).  sin2x+cos2 x  cos x  2cos 2 x  sinx  0.   1   b). 2sin  x    sin  2 x      3 6 2   d). sin 2 x  cos 2 x  3sin x  cos x 1  0. e). sin 2 x cos x  sin x cos x  cos 2 x  sinx  cos x f). 3 sin 2 x  3 sin x  cos 2 x  cos x  2 .   g). sin 2 x  cos x 2 x  4 2 sin  x    4 cos x  1  0 4 . 178. Lớp Toán Thầy-Diệp Tuân.  Tel: 0935.660.880.

<span class='text_page_counter'>(179)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. 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.......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 179. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(180)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. 4. Câu hỏi trắc nghiệm. Mức độ 3. Vận dụng Câu 74.(THPT Lê Hoàn-Thanh Hóa 2018) Tính tổng T tất cả các nghiệm của phương trình  2 cos x  1 sin 2 x  cos x   0   trên  0;  ta được kết quả là: sin x  1  2 2   A. T  . B. T  . C. T   . D. T  3 2 3 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 75.(Chuyên Hùng Vương Phú Thọ) Phương trình sin 2 x  3cos x  0 có bao nhiêu nghiệm trong khoảng  0;   A. 0 .. B. 1 .. C. 2 .. D. 3. Lời giải. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 76.(THPT Chuyên Vĩnh Phúc 2018) Tính tổng tất cả các nghiệm của phương trình sin 2 x  4sin x  2cos x  4  0 trong đoạn  0;100  của phương trình. A. 100 .. 180. B. 2476 .. Lớp Toán Thầy-Diệp Tuân. C. 25 .. D. 2475. Tel: 0935.660.880.

<span class='text_page_counter'>(181)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 77.(THPT Chuyên Lê Hồng Phong 2018) Tất cả các nghiệm của phương trình cos5x.cos x  cos 4 x là k k k A. x  B. x  C. x  k  k   . D. x  k  . k  . k   5 3 7 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 5  k 2 , k  . 6. cos x  3 sin x  0. 2sin x  1 5  k , k  . B. x   6.  k 2 , k  .. D. x . Câu 78.(THPT Sơn Tây-Hà Nội 2018) Giải phương trình A. x   C. x .  6. . 6.  k , k  .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 181. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(182)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 79.(THPT Chuyên Thái Bình 2018) cos x  sin 2 x  1  0 . Khẳng định nào dưới đây là đúng: Cho phương trình cos 3x A. Phương trình đã cho vô nghiệm. B. Nghiệm âm lớn nhất của phương trình là x  . . . 2 C. Phương trình tương đương với phương trình  sin x  1 2sin x  1  0 . D. Điều kiện xác định của phương trình là cos x  3  4 cos 2 x   0 .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 80.(THPT Chuyên Thái Bình 2018) Phương trình   khoảng  0,  ?  2 A. 1 .. B. 3 .. cos 4 x  tan 2 x có bao nhiêu nghiệm thuộc cos 2 x. C. 4 .. D. 2 .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 182. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(183)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. Câu 81.(SGD Vĩnh Phúc-KSCL 2018) Phương trình cos2 x  cos2 2 x  cos2 3x  cos2 4 x  2 tương đương với phương trình A. sin x.sin 2 x.sin 5x  0 . B. sin x.sin 2 x.sin 4 x  0 . C. cos x.cos 2 x.cos5x  0 . D. cos x.cos 2 x.cos 4 x  0 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 82.(THPT Lục Ngạn-Bắc Ninh 2018) Phương trình sin 5 x  sin 9 x  2sin 2 x  1  0 có một họ nghiệm là:  k 2  k 2  3    k A. x  . B. x  . C. x   k 2 . D. x  42 7 42 3 5 7 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 83.(THPT Kim Liên-Hà Nội 2018) Tìm tất cả các nghiệm của phương trình cos3x  sin 2 x  sin 4 x  0 ..  2 k , k . 6 3   5  k 2 , k  . C. x  k ; x   k 2 ; x  3 6 6 A. x .   k , k . 6 3    D. x   k ; x    k 2 , k  6 3 3 B. x . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 183. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(184)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... .................................................................................................................. Câu 84.(THTT Số 4-487 tháng 1 2018) Tổng tất cả các nghiệm của phương trình cos  sin x   1 trên  0; 2  bằng A. 0 .. B.  .. C. 2 .. D. 3. 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Câu 85.(THTT Số 4-487 2018) Xét phương trình sin 3x  3sin 2 x  cos 2 x  3sin x  3cos x  2 . Phương trình nào dưới đây tương đương với phương trình đã cho? A.  2sin x  1  2 cos 2 x  3cos x  1  0 . B.  2sin x  cos x  1 2cos x  1  0 . C.  2sin x  1 2cos x  1 cos x  1  0 .. D.  2sin x  1 cos x  1 2cos x  1  0 .. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 86.(THPT Lê Hoàn-Thanh Hóa 2018) Số vị trí điểm biểu diễn các nghiệm của phương trình sin 2 x  2 cos x  sin x  1  0 trên đường tròn lượng giác là: tan x  3 A. 4 . B. 1 . C. 2 . D. 3 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 184. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(185)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 87.(THPT Can Lộc-Hà Tĩnh-2018) 9  Số nghiệm của phương trình sin  2 x  2  A. 6 . B. 5 .. 15      3cos  x    1  2sin x với x   0; 2  là 2    C. 3 . D. 4 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 88.(THPT Lê Quý Đôn-Quãng Trị 2018) Giải phương trình: cos3x.tan 4 x  sin 5x . 2    3 A. x  k  , x   k . B. x  k 2 , x   k . 3 16 8 16 8     3 C. x  k , x   k . D. x  k , x   k 16 8 2 16 8 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 89.(THPT Trần Kỳ Phong 2018) Giải phương trình sin x.cos x  được số nghiệm là: A. 2016 nghiệm.. 185. B. 2017 nghiệm.. Lớp Toán Thầy-Diệp Tuân. 1 trên đoạn  ; 2018  ta 2. C. 2018 nghiệm.. D. 2019 nghiệm.. Tel: 0935.660.880.

<span class='text_page_counter'>(186)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. 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Câu 90.(THPT Chuyên Vĩnh Phúc 2018) Tìm số nghiệm của phương trình sin  cos x   0 trên đoạn x   0; 2  . A. 0 .. B. 1 .. C. 2 .. D. Vô số.. 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Câu 91.(THPT Yên Định 2018) Nghiệm của phương trình sin x  3 cos x  2sin 3 x là   2  2  k 2 , k  . A. x   k hoặc x   k , k . B. x   k 2 hoặc x  6 6 3 3 3  4    k 2 , k  . C. x    k 2 hoặc x  D. x   k , k  . 3 3 3 2 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 92.(THPT Lê Qúy Đôn-2018) Giải phương trình: cos3x.tan 4 x  sin 5x . 2    3 A. x  k  , x   k . B. x  k 2 , x   k . 3 16 8 16 8     3 C. x  k , x   k . D. x  k , x   k 16 8 2 16 8 Lời giải. 186. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 93.(Toán Học Tuổi Trẻ 484-10/2017) Tính tổng S các nghiệm của phương trình.  2 cos 2 x  5  sin 4 x  cos 4 x   3  0. trong khoảng.  0; 2  . A. S . 11 . 6. B. S  4 .. C. S  5 .. D. S . 7 6. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 94.(THTT Số 2-485 tháng 11 2018) Số nghiệm của phương trình cos 2 x  2cos 3x.sin x  2  0 trong khoảng  0;   là A. 0 .. B. 1 .. C. 2 . D. 3 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 187. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 95.(THPT Bình Xuyên 2018) Phương trình đương với phương trình nào sau đây: A.  sin x  sin 2 x  sin 3x  cos x  cos 2 x   0 . C.  sin x  sin 2 x  sin 3x  sin x  sin 2 x   0 ..  sin x  sin 2 x sin x  sin 2 x   sin 2 3x. tương. B.  sin x  sin 3x  sin x  0 . D.  sin x  sin 3x  sin 3x  0. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................  3  ;    của phương trình Câu 96.(THTT Số 3-486 tháng 12 năm 2018) Tìm số nghiệm thuộc   2   3  3 sin x  cos   2x  .  2  A. 0 . B. 1 . C. 2 . D. 3 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................    Câu 97.(THPT Triệu Sơn 1-lần 1 2018) Số nghiệm nằm trong đoạn   ;  của phương trình  2 2 sin 5x  sin 3x  sin 4 x là A. 5 . B. 7 . C. 9 . D. 3 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 188. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................  4   Câu 98.(THPT Chuyên ĐHSP-Hà Nội 2018) Số nghiệm thuộc khoảng   ;  của phương trình  3 2   cos   x   3 sin x  sin  3 x   là 2  A. 4 . B. 3 . C. 6 . D. 2 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Câu 99.(THPT Kinh Môn 2 Hải Dương 2018) Phương trình lượng giác: cos3x  cos 2 x  9sin x  4  0 trên khoảng  0;3  . Tổng số nghiệm của phương trình trên là: 25 A. . 6. B. 6 .. C. Kết quả khác.. D.. 11 3. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... 189. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Câu 100.(THPT Lê Quý Đôn-Hải Phòng 2018) Biểu diễn tập nghiệm của phương trình cos x  cos 2 x  cos3x  0 trên đường tròn lượng giác ta được số điểm cuối là A. 6 . B. 5 . C. 4 . D. 2 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 101.(THPT Yên Lạc Vĩnh Phúc 2018) Tập tất cả các nghiệm của phương trình sin 2 x  2sin 2 x  6sin x  2cos x  4  0 là A. x   C. x . 190.  2. . 3.  k 2 , k  ..  k 2 , k  .. Lớp Toán Thầy-Diệp Tuân. B. x   D. x .  2. . 2.  k 2 , k  ..  k , k . Tel: 0935.660.880.

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Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 102.(THPT Kinh Môn 2 2018) Phương trình lượng giác: cos3x  cos 2 x  9sin x  4  0 trên khoảng  0;3  . Tổng số nghiệm của phương trình trên là: A.. 25 . 6. B. 6 .. C. Kết quả khác.. D.. 11 3. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 103.(THPT Lê Xoay 2018).   Số nghiệm của phương trình sin 2 x  cos x  1  log 2  sin x  trên khoảng  0;  là:  2 A. 4 . B. 3 . C. 2 . D. 1 Lời giải. 191. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Mức độ 4. Vận dụng cao Câu 104.(THPT Cổ Loa-Hà Nội-lần 2018) Tìm tất cả các giá trị của tham số m để phương trình    cos 4 x  cos 2 3x  m sin 2 x có nghiệm x   0;  .  12  1  1 1   A. m   0;  . B. m   ; 2  . C. m   0;1 . D. m   1;  . 4  2 2   Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 105.(THPT Hai Bà Trưng-Vĩnh Phúc 2018) Tổng các nghiệm của phương trình 2cos3x  2cos 2 x  1  1 trên đoạn  4 ;6  là: A. 61 .. B. 72 .. C. 50 .. D. 56. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... 192. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 106.(THPT Kinh Môn-Hải Dương 2018) Cho phương trình sin 2018 x  cos 2018 x  2  sin 2020 x  cos 2020 x  . Tính tổng các nghiệm của phương trình trong khoảng  0; 2018 2.  1285  A.   .  4 . B.  643  . 2. C.  642   . 2. 2.  1285  D.     2 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(194)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. Câu 107.(THPT Chuyên Hạ Long 2018) Cho phương trình sin 2 x.tan x  cos 2 x.cot x  2sin x cos x  nghiệm dương nhỏ nhất của phương trình. 3 5 A.  . B. . 2 6. 4 3 . Tính hiệu nghiệm âm lớn nhất và 3. C. . 5 . 6. D. . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 108.(Sở GD & ĐT Vĩnh Phúc 2018) Có bao nhiêu giá trị nguyên dương của m để phương trình sin 2 x  2sin x  cos x  cos 2 x  m sin 2 x có nhiều hơn một nghiệm trong khoảng  0; 2π  ? A. 3 .. B. 2 .. C. 4 . D. 5 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Câu 109.(THPT Chuyên Tiền Giang 2018) Tìm tất cả các giá trị của m để phương trình sin 4 x  cos 4 x  cos 2 4 x  m có bốn nghiệm phân biệt    thuộc đoạn   ;  .  4 4 47 3 47 3 47 3 47 3 m . m . m A. m  hoặc m  . B. C. D. 64 2 64 2 64 2 64 2. 194. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Câu 110.(THPT Kinh Môn 2018) Cho phương trình sin 2018 x  cos 2018 x  2  sin 2020 x  cos 2020 x  . Tính tổng các nghiệm của phương trình trong khoảng  0; 2018  2.  1285  A.   .  4 . B.  643  . 2. C.  642   . 2. 2.  1285  D.   .  2 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 195. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(196)</span> Trung Tâm Luyện Thi Đại Học Amsterdam. Chương I-Bài 3. Một Số Phương Trình Lượng Giác. .......................................................................................................................................................................................................... .................................................................................................................. Câu 111.(THPT Trần Nhân Tông 2018) Số nghiệm của phương trình: sin 2015 x  cos 2016 x  2  sin 2017 x  cos 2018 x   cos 2 x trên  10;30 là: A. 46 .. B. 51 .. C. 50 .. D. 44. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(197)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học. §BÀI 4.. LƯỢNG GIÁC TRONG CÁC ĐỀ THI ĐẠI HỌC 2002-2015. Bài tập 1. Giải các phương trình lượng giác sau: cos 3 x  sin 3 x   a). 5  sin x    cos 2 x  3, x  (0; 2 ). 1  2sin 2 x   b). sin 2 3x  cos 2 4 x  sin 2 5 x  cos 2 6 x. c). cos3x  4cos 2 x  3cos x  4  0, x  0; 14.. (ĐH khối A năm 2002) (ĐH khối B năm 2002) (ĐH khối D năm 2002). Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(198)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học. .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 2. Giải các phương trình lượng giác sau: cos 2 x 1  sin 2 x  sin 2 x. a). cot x  1  (ĐH khối A năm 2003 1  tan x 2 2  b). cot x  tan x  4sin 2 x  (ĐH khối B năm 2003) sin 2 x x x  c). sin 2    tan 2 x  cos 2  0. (ĐH khối D năm 2003) 2 2 4 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(199)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học. .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 3. Giải các phương trình lượng giác sau: a). 5sin x  2  3(1  sin x) tan 2 x. (ĐH khối B năm 2004) b). (2cos x  1)(2sin x  cos x)  sin 2 x  sin x. (ĐH khối D năm 2004) Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Bài tập 4. Giải phương trình lượng giác sau: cos 2 3x cos 2 x  cos 2 x  0. (ĐH khối A năm 2005). Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 199. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(200)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học Bài tập 5. Giải phương trình lượng giác sau: 1  sin x  cos x  sin 2 x  cos 2 x  0. (ĐH khối B năm 2005). Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................    3  Bài tập 6. Giải phương trình lượng giác cos 4 x  sin 4 x  cos  x   sin  3 x     0. 4  4 2  (ĐH khối D năm 2005) Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 2(cos6 x  sin 6 x)  sin x cos x  0. 2  2sin x (ĐH khối A năm 2006) Lời giải .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 7. Giải các phương trình lượng giác sau:. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 200. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(201)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... x  Bài tập 8. Giải các phương trình lượng giác sau: cot x  sin x 1  tan x tan   4. 2  (ĐH khối B năm 2006) Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 9. Giải các phương trình lượng giác sau: cos3x  cos 2 x  cos x 1  0. (ĐH khối D năm 2006) Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 10. Giải các phương trình lượng giác sau: (1  sin 2 x) cos x  (1  cos 2 x) sin x  1  sin 2 x. (ĐH khối A năm 2007) Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 201. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(202)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 11. Giải các phương trình lượng giác sau: 2sin 2 2 x  sin 7 x  1  sin x. (ĐH khối B năm 2007) Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. 2. x x  Bài tập 12. Giải các phương trình lượng giác sau:  sin  cos   3 cos x  2. 2 2  (ĐH khối D năm 2007) Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 13. Giải các phương trình lượng giác sau: 1 1  7    4sin   x . a). 3  sin x   4  sin  x   2   b). sin 3 x  3 cos3 x  sin x cos 2 x  3 sin 2 x cos x. c). 2sin x 1  cos 2 x   sin 2 x  1  2cos x.. (ĐH khối A năm 2008). (ĐH khối B năm 2008) (ĐH khối D năm 2008). Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 202. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(203)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(204)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 14. Giải các phương trình lượng giác sau (1  2sin x) cos x a).  3. (1  2sin x)(1  sin x) b). sin x  cos x sin 2 x  3 cos3x  2(cos 4 x  sin 3 x). c).. 3 cos 5 x  2sin 3 x cos 2 x  sin x  0.. (ĐH khối A năm 2009) (ĐH khối B năm 2009) (ĐH khối D năm 2009). Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(205)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học. Bài tập 15. Giải các phương trình lượng giác sau :   (1  sin x  cos 2 x) sin  x   1 4   cos x. a). (ĐH khối A năm 2010) 1  tan x 2 b). (sin 2 x  cos 2 x) cos x  2cos 2 x  sin x  0. (ĐH khối B năm 2010) c). sin 2 x  cos 2 x  3sin x  cos x 1  0. (ĐH khối D năm 2010) Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(206)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 16. Giải các phương trình lượng giác sau: 1  sin 2 x  cos 2 x  2 sin x sin 2 x. a). (ĐH khối A năm 2011) 1  cot 2 x b). sin 2 x cos x  sin x cos x  cos 2 x  sin x  cos x. (ĐH khối B năm 2011) sin 2 x  2 cos x  sin x  1 c). (ĐH khối D năm 2011)  0. tan x  3 Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... 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................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 206. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(207)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 17. Giải các phương trình lượng giác sau: a). 3 sin 2 x  cos 2 x  2 cos x  1. b). 2(cos x  3 sin x) cos x  cos x  3 sin x  1.. (ĐH khối A năm 2012) (ĐH khối B năm 2012). c). sin 3x  cos 3x  sin x  cos x  2 cos 2 x. (ĐH khối D năm 2012) Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Bài tập 18. Giải các phương trình lượng giác sau:   a). 1  tan x  2 2 sin  x    4  2 b). sin 5 x  2cos x  1. c). sin 3x  cos 2 x  sin x  0. Lời giải. 207. Lớp Toán Thầy-Diệp Tuân. (ĐH khối A năm 2013) (ĐH khối B năm 2013) (ĐH khối D năm 2013). Tel: 0935.660.880.

<span class='text_page_counter'>(208)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Bài tập 19. Giải các phương trình lượng giác sau sin x  4cos x  2  sin 2 x. (ĐH khối A năm 2014). Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 2(sin x  2cos x)  2  sin 2 x. (ĐH khối B năm 2014) Lời giải .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 20. Giải các phương trình lượng giác sau. .......................................................................................................................................................................................................... ................................................................................................................... 208. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thi Đại Học. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 21. Giải phương trình: 2sin 2 x  7sin x  4  0. (TN THPT QG năm 2016) Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 22. Giải các phương trình lượng giác sau: cos x cos3x  sin 2 x sin 6 x  sin 4 x sin 6 x  0. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 1 Bài tập 23. Giải các phương trình lượng giác sau: cos x cos 2 x cos 3x  sin x sin 2 x sin 3x   2. 209. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(210)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Bài tập 24. Giải các phương trình lượng giác sau: cot x  cos 2 x  sin x  sin 2 x cot x  cos x cot x. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 25. Giải các phương trình lượng giác sau sin 2 x cos x  sin x cos x  cos 2 x  sin x  cos x. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 210. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thi Đại Học. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 26. Giải các phương trình lượng giác sau 4  3sin x  sin 3 x  3cos 2 x  cos 6 x. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 27. Giải các phương trình lượng giác sau 2sin 3 x  cos 2 x  cos x  0. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 211. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(212)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học. .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 28. Giải các phương trình lượng giác sau 2cos x cos 2 x cos3x  5  7cos 2 x. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... Bài tập 29. Giải các phương trình lượng giác sau sin 2 x(4 cos 2 x  1)  cos x(sin x  cos x  sin 3 x). Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Bài tập 30. Giải các phương trình lượng giác sau cos x  3(sin 2 x  sin x)  4cos 2 x cos x  2cos 2 x  2  0. Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 212. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thi Đại Học. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 31. Giải các phương trình lượng giác sau (sin x  cos x) 2  2sin 2 x 2       sin   x   sin   3x     2 1  cot x 2  4  4 . Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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.......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 213. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

<span class='text_page_counter'>(214)</span> Trung Tâm Luyện Thi Amsterdam Chương I-Bài 4. Một Số Phương Trình Lượng Giác Thi Đại Học. 1 1 15cos 4 x    2 2 2cot x  1 2 tan x  1 8  sin 2 2 x Lời giải .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 32. Giải các phương trình lượng giác sau. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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.......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ...................................................................................................................   2 sin  x    4    cos 3 x  2 sin  2 x    1. Bài tập 33. Giải các phương trình lượng giác sau tan x  1 4  Lời giải. 214. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Một Số Phương Trình Lượng Giác Thi Đại Học. .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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Bài tập 34. Giải các phương trình lượng giác sau  3    3sin 2 x cos   x   sin 2   x  cos x  sin x cos 2 x  3sin 2 x cos x.  2  2  Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... .................................................................................................................. Bài tập 35. Giải các phương trình lượng giác sau (2sin x  1)(cos 2 x  sin x)  2sin 3 x  6sin x  1  2 cos x  3  0. 2 cos x  3. 215. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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Lời giải .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 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.......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... .......................................................................................................................................................................................................... ................................................................................................................... 216. Lớp Toán Thầy-Diệp Tuân. Tel: 0935.660.880.

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