TAI LIEU HOC TAP LG 10 FULL TRAN TAI

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TÀI LIỆU HỌC TẬP TOÁN 10 – CHẤT LƯỢNG

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

PHẦN V: BIẾN ĐỔI LƯỢNG GIÁC
A – HỆ THỨC LƯỢNG GIÁC CƠ BẢN
 tan  

sin 
.
cos 

 cot  

 tan .cot   1 .

 sin 2   cos2   1 .
 1  tan2  

cos 
.
sin 

1
.
cos2 

 1  cot2  

1
.

sin2 

BÀI TẬP ÁP DỤNG
Bài1.

Chứng minh các đẳng thức sau
1/

cos2 x  sin2 x  1  2 sin2 x .

2/

2 cos2 x  1  1  2 sin2 x .

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3/

3  4 sin2 x  4 cos2 x  1 .

4/

sin x cot x  cos x tan x  sin x  cos x .

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5/

sin 4 x  cos 4 x  1  2 sin 2 x cos2 x .

6/

cos4 x  sin4 x  cos2 x  sin2 x .

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7/

4 cos2 x  3  1  2 sin x 1  2 sin x  . 8/

1  cos xsin

2



x  cos x  cos2 x  sin2 x .

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9/

sin4 x  cos4 x  1  2cos2 x  2sin2 x 1 . 10/ sin 3 x cos x  sin x cos 3 x  sin x cos x .

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11/ tan2 x  sin2 x  tan2 x sin2 x .

12/ cot2 x  cos2 x  cot2 x cos2 x .

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Bài2.

Chứng minh các đẳng thức sau

1/

tan x  cot x 

1
.
sin x cos x

2/

1  cos x
sin x

.
sin x
1  cos x

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3/

1
1

 1.
1  tan x 1  cot x

4/




1  1  1  1   tan2 x  0 .
 

cos x 
cos x 

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5/

1  sin2 x
 1  2 tan2 x .
2
1  sin x

6/

tan x tan y 

tan x  tan y
.
cot x  cot y

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BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

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7/

1  cot4 x 

2
1

.
2
sin x sin 4 x

8/

tan x 

cos x
1

.
1  sin x
cos x

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9/

1

1  cos x1  cot x  1  cos x .
2

10/

1  cos x 1  cos x
4 cot x


.
1  cos x 1  cos x
sin x

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sin2 x  2 cos x  1
cos x

13/
.
2
1  cos x
2  cos x  cos x

14/

sin x  cos x  1
2 cos x

.
1  cos x
sin x  cos x  1

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Bài3.

Chứng minh các đẳng thức sau

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1/

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

sin 6 x  cos6 x  1  3 sin2 x cos2 x .

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2/







sin 6 x  cos6 x  sin2 x  cos2 x 1  sin2 x cos2 x .

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3/





2

sin 8 x  cos 8 x  1  2 sin2 x cos 2 x  2 sin 4 x cos 4 x .

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4/







sin 8 x  cos8 x  sin2 x  cos2 x 1  2 sin2 x cos2 x .

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Bài4.

Chứng minh các đẳng thức sau

1/

1  sin x  cos x  tan x  1  cos x1  tan x .

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2/

1  tan x1  cot x sin x cos x  1  2 sin x cos x .

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BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

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3/

1  tan x cos

2

x  1  cot x  sin 2 x  sin x  cos x  .
2

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4/

sin 2 x tan x  cos2 x cot x  2 sin x cos x  tan x  cot x .

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5/

sin 2 x tan 2 x  4 sin2 x  tan 2 x  3 cos2 x  3 .

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Bài5.

Chứng minh các đẳng thức sau
1/

sin2   cos2 
tan   1

.
1  2 sin  cos 
tan   1

2/

tan2   sin2 
 tan 6  .
2
2
cot   cos 

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3/

cos2 
sin2 

 1  sin  cos  .

1  tan  1  cot 

4/

tan   sin 
1

.
3
sin 
cos  1  cos 

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5/

1
 tan2   cot2   2 .

2
2
sin  cos 

6/

1
3 tan2 
2

tan


 1.
cos2 
cos2 

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7/

tan2   tan2  sin2   sin 2 
.

tan2  tan2 
sin2  sin2 

8/

2



1  cos   1  cos  

  2 cot  .

1

sin   sin2 



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9/

1

1  cos  1  cot   1  cos  .
2

10/ 1  tan   tan2   tan3  

sin   cos 
.
cos3 

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11/

sin 
cos 
1  cot2 


.
sin   cos  cos   sin 1  cot2 

12/

tan2  1  cot2 
1  tan4 
.

.
1  tan2  cot2 
tan2   cot2 

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
  4 cot  .
 1  cos 
1  cos  

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Bài6.

Rút gọn các biểu thức sau
1/ P1  sin 4   sin2  cos2  .

2/ P2  sin 4   cos4   cos2  .

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3/ P3  sin2   sin2  cot2  .

4/ P4  cos2   cos2  cot2  .

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



5/ P5  1  sin2  cot2   1  cot2  .

6/ P6  sin2  tan  cos2  cot  2sin cos  .

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7/

P7 

2 cos2   1
.
sin   cos 

8/

P8 

1  cos 
2

sin 



1
.
1  cos 

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9/

P9 

cot   cot 
.
tan   tan 

10/ P10  tan  

cos 
.
1  sin 

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11/ P11 

sin x  tan x
 sin x cot x .
tan x

12/ P12 

cos x tan x
sin2 x

 cot x cos x .

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13/ P13 

cos2 x  cot2 x
2

2

sin x  tan x

14/ P14 

.

1  sin2 x cos2 x
2

cos x

 cos2 x .

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sin3 x  sin x cos2 x  cos x
15/ P15 
.
1  2 sin x cos x

2

16/ P16

sin x  cos x



1

tan x  sin x cos x

.

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17/ P17 

sin2 x  sin2 x tan2 x
2

2

2

cos x  cos x tan x

.

18/ P18 

cot2 x  cos2 x
2

cot x

 1  cos2 x .

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19/ P19 

cos2 x  sin2 y
2

2

sin x sin y

 cot2 x cot2 y .

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

20/ P20 

1
sin x
cos x cot x


.
sin x 1  cot x
1  tan x

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21/ P21 


1  cos x 1  sin x
1 
1  cotx  1  .

 tan x  cotx . 22/ P22  1  cotx 


sin x
cos x
sin x 

sin x 

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23/ P23  1  tanx cos2 x 1  cotx sin2 x .

24/ P24 



, 0  x  .
2
sinx  cot2 x  cos2 x 
1

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Bài7.

Biến đổi các biểu thức sau thành tích số
1/

A  2 cos2 x  1 .

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2/

B  3  4 sin2 x .

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3/

C  sin x cos x  cos2 x  1 .

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4/

D  sin2 x  sin x cos x  1 .

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5/

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

E  1  sin x  cos x  tan x .

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6/

F  tan x  cot x  sin x  cos x .

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7/

G  cos x tan2 x  1  cos x  .

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8/





H  3  4 cos2 x  sin x 2 sin x  1 .

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9/

I  sin2 x  3 cos2 x  6 cos x  2 sin x .

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10/ J  cos3 x  sin3 x  sin x  cos x .
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11/ K  cos3 x  cos2 x  2 sin x  2 .
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12/ L  cos2 x  sin3 x  cos x .
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13/ M  1  cos x  cos2 x  sin x 1  cos x  .
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14/ N  2 cos3 x  2 cos2 x  sin x  1 .
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15/ O  cos3 x  sin3 x  2 sin2 x  1 .
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16/ P  2 sin x  12 cos x  2 sin x  1  3  4 cos2 x .
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17/ Q  2 cos x  1sin x  cos x   1 .
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18/ R  4 sin3 x  3 cos3 x  3 sin x  sin2 x cos x .
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19/ S  1  sin x  tan2 x  1  cos x .

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20/ T  2  5 sin x  3 1  sin x  tan2 x .
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21/ U  2 sin x cos x  2 sin2 x  3 sin x  cos x  1 .
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



22/ V  tan x  3 cot x  4 sin x  3 cos x .
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23/ X  3 sin x  2 cos x  3 tan x  2 .
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24/ Y  2  tan x  sin x   3 cot x  cos x   5 .
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25/ Z  3 cot x  cos x  5 tan x  sin x  2 .
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Bài8.

Chứng minh các biểu thức sau không phụ thuộc vào biến x
1/

A  cos4 x  sin4 x  2 sin2 x .

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2/

B  sin4 x  sin2 x cos2 x  cos2 x .

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3/

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

C  cos4 x  sin2 x cos2 x  sin2 x .

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4/









D  cos4 x 2 cos2 x  3  sin 4 x 2 sin2 x  3 .

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5/

E  sin6 x  cos6 x  2 sin4 x  cos4 x  sin2 x .

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6/

F  sin x.

1
1

,
1  cos x 1  cos x


0  x 


 
.
4 

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7/

G  sin 4 x  4 cos2 x  cos 4 x  4 sin2 x .

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8/

H  cos2 x cot2 x  5 cos2 x  cot2 x  4 sin2 x .

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9/

I  1  cot x  sin 3 x  1  tan x cos3 x  sin x  cos x .

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





10/ J  sin4 x  cos 4 x  1 tan2 x  cot2 x  2 .

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...................................................................................................................................................
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

 



11/ K  3 sin8 x  cos 8 x  4 cos6 x  2 sin 6 x  6 sin 4 x .
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................









12/ L  sin4 x 1  sin2 x  cos 4 x 1  cos2 x  5 sin 2 x cos2 x  1 .
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................



 
2



13/ M  2 sin4 x  cos4 x  sin2 x cos2 x  sin 8 x  cos8 x .
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Bài9.

Chứng minh các biểu thức sau không phụ thuộc vào biến x

1/

A

2
cot x  1

.
tan x  1 cot x  1

...................................................................................................................................................
...................................................................................................................................................

2/

B



1  tan2 x
2

4 tan x



2



1
2

4 sin x cos2 x

.

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3/

C

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

1  tan2 x
 1  tan2 x 1  cot2 x .
tan x







...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................

1  cot x
D

2

2

4/

cot2 x



1
sin2 x cos2 x

.

...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................

...................................................................................................................................................
5/

E

1  sin6 x
cos6 x



3 tan2 x
cos2 x

.

...................................................................................................................................................
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6/

F

tan2 x  cos2 x
sin2 x



cot2 x  sin2 x
cos2 x

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sin4 x  cos4 x  1
sin6 x  cos6 x

.

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B – ĐƯỜNG TRÒN LƯỢNG GIÁC – CUNG GÓC LƯỢNG GIÁC
CUNG LIÊN KẾT

1. Đường tròn lượng giác và dấu góc lượng giác
sinx

Cung

1

phần tư

π

I

II III IV

+

+

–

–

+

–

–

+

+

–

+

–

+

–

+

–

2

Giá trị LG
(II)

(I)

π

0 1
2π

O

-1

cosx

(IV)

(III)
3π
2

(Nhất cả – Nhì sin – Tam tan – Tứ cos)

-1

2.Cung liên kết

Cung đối nhau

Cung bù nhau

cos(a)  cos a

sin(   a)  sin a



sin   a   cos a
2


sin(a)   sin a

cos(   a)   cos a



cos   a   sin a
2


tan(a)   tan a

tan(   a)   tan a



tan   a   cot a
2


cot(a)   cot a

cot(   a)   cot a



cot   a  tan a
 2


Cung hơn kém

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

Cung phụ nhau

Cung hơn kém

sin(   a)   sin a



sin   a   cos a
2


cos(   a)   cos a



cos   a    sin a
2



tan(   a)  tan a



tan   a    cot a
2



cot(   a)  cot a



cot   a    tan a
2



2

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BÀI TẬP ÁP DỤNG
MỐI LIÊN HỆ GIỮA ĐƠN VỊ ĐỘ VÀ RADIAN
Bài10.

Đổi số đo các cung sau ra độ, phút, giây:

2 3 5  7 13 3
15 
,
,
,
,
,
,  , 2,
, .
3
4
9
5
3
3
4
7 15

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...................................................................................................................................................
Bài11.

Đổi số đo các góc sau ra radian rad :
900, 360, 150 , 720 , 2700, 2400, 5400, 7500 , 210 .

...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................

...................................................................................................................................................
Bài12.

Đổi số đo độ của các cung tròn sau thành số đo radian (chính xác đến phần nghìn):
21030 ', 75054 ', 12036 ' , 15030 ',  105045 ' 30 '', 27 038 ' 49 '' .

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Bài13.

Điền các giá trị thích hợp vào ô trống

00

150

300

450

600

750

900

1200

1500

1800


16


8


6

5
12


2

2
3

3
4

5
6



4
3

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Radian

7
4


12

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC



5
6

Độ


2

2
3

3
4

1350

500

8100

TÍNH ĐỘ DÀI CUNG TRÒN
Bài14.

Một đường tròn có bán kính 10 cm  . Tính độ dài các cung có số đo là
2  3 7 13
3
3
,
,
,
,
,  2,
, 450, rad, 75 0 .
3 18 5
3
6
4
2

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Bài15.

Kim giờ của đồng hồ dài 7 cm  , kim phút dài 10 cm  . Tính quãng đường kim phút,
kim giờ đi được trong 30 phút.

...................................................................................................................................................
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...................................................................................................................................................
Bài16.

Bánh xe có đường kính (tính cả lốp) là 55 cm . Nếu xe chạy với vận tốc 40 km /h
thì trong một giây bánh xe quay được bao nhiều vòng ?

...................................................................................................................................................
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Bài17.

Kim phút và kim giờ của đồng hồ lớn nhà Bưu Điện Hà Nội theo thứ tự dài
1, 75 m  và 1, 26 m  . Hỏi trong 15 phút, mũi kim phút và mũi kim giờ vạch trên

cung tròn có độ dài bằng bao nhiêu mét ?
...................................................................................................................................................
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BIỂU DIỄN NGỌN CUNG LƯỢNG GIÁC
Bài18.

Biểu diễn các góc sau lên đường tròn lượng giác gốc A:
300 ,  450, 1200,  1200, 3300,
6300, 7500 ,  12500,

7
4
15 2010
, 
,  750 0, 11250 ,
,
.

4
3
2
4

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Bài19.




Xác định điểm cuối của cung có số đo: k2, k, k , k , k , k   .
2
3
4

...................................................................................................................................................
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Bài20.

Xác định điểm ngọn của các họ nghiệm sau đây trên đường tròn lượng giác với
k  .

1/ x  k .

2/ x 


 k2 .
2

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3/ x  4k .

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

4/ x 


 k .
3

...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
5/ x  


 k2 .
6

6/ x  


 k .
4

...................................................................................................................................................
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...................................................................................................................................................
...................................................................................................................................................
7/ x 

k
.
3

8/ x 

 k2

.
3
3

...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
9/ x 

 k3

.
4
4

10/ x 

 k

.
6
3

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...................................................................................................................................................

...................................................................................................................................................
Bài21.

   và sđ AN
  k với
Cho hai điểm M và N sao cho sđ AM
k   . Tìm k   để:
6
798

1/ M trùng với N.

2/ M đối xứng với N qua tâm O.

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Bài22.

Hai góc lượng giác có số đo radian

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

35
m
và
với m là số nguyên có thể có cùng
3
5

tia đầu và tia cuối hay không ?
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...................................................................................................................................................

GÍA TRỊ LƯỢNG GIÁC CỦA MỘT CUNG (GÓC)
Bài23.

Xác định dấu của các giá trị lượng giác hoặc biểu thức
1/

sin x, cos x, tan x, cot x với   x 

3
.
2

...................................................................................................................................................





A  sin 400.cos 2900 .

4/





B  sin 250 .cos 1700 .

...................................................................................................................................................
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...................................................................................................................................................
5/





C  sin 2250. tan1300.cot 1750 .

6/





D  cos1950. tan 2690.cot 980 .

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7/





E  sin 500.cos 3000 .

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

8/

F  sin 2150. tan

21
.
7

...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
9/

G  cot

 2 
3
. sin   .
 3
5

10/ H  cos

4

4
9
. sin . tan
.cot
.
5
3
3

5

...................................................................................................................................................
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...................................................................................................................................................
Bài24.

Tính giá trị còn lại của góc x, biết
1/

sin x 

1
với 900  x  1800 .
2

2/

sin x  

4
với 2700  x  3600 .
5

...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
3/

sin x  

3
3
với   x 
.
5
2

4/

cos x 

1

với 0  x  .
4
2

...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
5/

cos x 

3
với 0  x  900 .

5

6/

cos x  

5
với 1800  x  2700 .
13

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...................................................................................................................................................
...................................................................................................................................................
7/

cos x 

22 | THBTN – CA

2
5

với 


 x  0.
2

8/

cos x 

4
với 2700  x  3600 .
5

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...................................................................................................................................................
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9/

sin x 

5

với
 x  .
13
2

10/ sin x  

1
với 1800  x  2700 .
3

...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
11/ tan x  3 với   x 

3
.
2

12/ tan x  2 với


 x  .
2

...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
13/ tan x  

1

với  x   .

2
2

14/ cot x  3 với   x 

3
.
2

...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
15/ tan x 

3
3
với   x 
.
4
2

16/ tan x   2 với


 x  .
2

...................................................................................................................................................
...................................................................................................................................................

3
 x  2 .
2

20/ cot150  2  3

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Bài25.

Tính giá trị của các biểu thức lượng giác sau
1/ Cho tan x  2 . Tính: A1 

5 cot x  4 tan x
2 sin x  cos x
, A2 
.
5 cot x  4 tan x
cos x  3 sin x

...................................................................................................................................................
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...................................................................................................................................................

...................................................................................................................................................
2/ Cho cot x  2 . Tính: B1 

3 sin x  cos x
sin x  3 cos x
, B2 
.
sin x  cos x
sin x  3 cos x

...................................................................................................................................................
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...................................................................................................................................................
...................................................................................................................................................
3/ Cho cot x  2 . Tính: C1 

2 sin x  3 cos x
2
, C2 
.
2
3 sin x  2 cos x
cos x  sin x cos x

...................................................................................................................................................
...................................................................................................................................................

D

,
D

,
4
5
sin x  sin2 x cos x
sin 3 x  2 cos3 x
2 cos x  sin3 x

...................................................................................................................................................
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...................................................................................................................................................
5/ Cho sin x 

3

cot x  tan x
, 0  x  . Tính E 

.
5
2
cot x  tan x

...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
6/ Cho sin x 

8 tan2 x  3 cot x  1
1
, 900  x  1800 . Tính F 
.
tan x  cot x
3

...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................

cot x  3 tan x
2
7/ Cho cos x   . Tính G 
.
2 cot x  tan x
3

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