ON THI DAI HOC

BÀI TẬP LƯỢNG GIÁC
65. 2sin x cos 2 x + sin 2 x cos x = sin 4 x cos x
2
66. tan x + cos x − cos x = sin x ( 1 + tan x tan 2x )

1. tan x = cot x + 4 cos 2 x
2

π
π
2


2. sin  2 x − ÷ = sin  x − ÷+
4
4 2


π

π

67. (2sin x − 1)(2 cos 2 x + 2 sin x + 1) = 3 − 4 cos2 x
68. sin 3 x + cos 3 x = 3. sin x. cos x − 1





3. 2sin  x + ÷− sin  2 x − ÷ =
3

6
2
1





sin 2 x cos 2 x
+
= tan x − cot x
4.
cos x
sin x

69. tan 2 x −

π
70. (1 + 2 cos 3x). sin x + sin 2 x = 2 sin 2 (2 x + )
4

π

5. 2 2 sin  x − ÷cos x = 1
 12 
6. (1 − tan x)(1 + sin 2 x) = 1 + tan x

7. cos 3 x cos 3 x − sin 3 x sin 3 x =

2+3 2

8

π




8. 2sin  2 x − ÷+ 4sin x + 1 = 0
6


π

π





9. sin  − ÷− cos  − ÷ = 2 cos
2
 2 4
2 4
5x

x

tan x
=2
cot 3 x

3x

10. (2sin 2 x − 1) tan 2 2 x + 3(2 cos 2 x − 1) = 0
11. cos 2 x + ( 1 + 2 cos x ) ( sin x − cos x ) = 0
12. cos3 x + sin 3 x + 2sin 2 x = 1
2 x
13. 3sin x + cos 2 x + sin 2 x = 4sin x cos
2
4
4
14. 4(sin x + cos x) + cos 4 x + sin 2 x = 0
1
1
−
= 2 cot 2 x
15. sin 2 x + sin x −
2sin x sin 2 x
16. 2 cos2 x + 2 3 sin x cos x + 1 = 3(sin x + 3 cos x)

71. (2 cos x − 1)(sin x + cos x) = 1
x
72. cot x + sin x(1 + tan x. tan ) = 4
2
73. 2 sin 3 x(1 − 4 sin 2 x) = 1
74. (1 − tan x)(1 + sin 2 x ) = 1 + tan x
75. cos 2 x + 5 = 2(2 − cos x)(sin x − cos x)
76. 2 cos 2 2 x + cos 2 x = 4 sin 2 2 x. cos 2 x
77. 3 tan 2 x + 3 cot 2 x + 4 tan x + 4 cot x + 2 = 0
78. (cos 2 x − cos 4 x) 2 = 6 + 2 sin 3x

79. sin 3 x(cos x − 2 sin 3 x) + cos 3 x(1 + sin x − 2 cos 3 x) = 0
80. sin 2 x + cos 2 x + tan x = 2
81. 2 tan x + cot 2 x = 2 sin 2 x +

1
sin 2 x

82. 2 sin x + cot x = 2 sin 2 x + 1
83. 1 + 3 tan x = 2 sin 2 x
84. tan x + 2 cot 2 x = sin 2 x
85. 4 cos 2 x − cos 3 x = 6 cos x − 2(1 + cos 2 x)
86. 3 sin x − 2 cos x = 2 + 3 tan x
87. sin 3x = cos 2 x. cos x(tan 2 x + tan 2 x)
88. 2 cos 2 x + 2 cos 2 2 x + 2 cos 2 3 x = cos 4 x( 2 sin 2 x + 1) + 3
1 − cos 2 x
sin 2 2 x

17. 4sin 3 x + 4sin 2 x + 3sin 2 x + 6 cos x = 0
3π 
2 x
2
18. 4sin − 3 cos 2 x = 1 + 2 cos  x − ÷.

89. 1 + cot 2 x =

π
3
19. 2 2 cos  x − ÷− 3cos x − sin x = 0

91. sin 3 x + cos 2 x = 1 + 2 sin x. cos 2 x



2



4 

4

cos 2 x − 1
π

2
20. tan  + x ÷− 3 tan x =
cos 2 x
2

21. 1 − sin x + 1 − cos x = 1
π
1
1

=
22. 2 2 cos  x + ÷+

4  sin x cos x

23. sin 4 x sin 7 x = cos 3 x cos 6 x

24. sin x + sin 2 x = 3 ( cos x + cos 2 x )

2
25. cos 2 x + cos x ( 2 tan x − 1) = 2

26. 3 − tan x ( tan x + 2sin x ) + 6 cos x = 0
27. 3cos 4 x − 8cos6 x + 2 cos 2 x + 3 = 0

π
4

90. sin 4 x − cos 4 x = 1 + 4 2 sin( x − )
3x
−2=0
4
93. sin x. sin 2 x + sin 3x = 6 cos 3 x

92. cos 2 x + cos

2
94. sin 2 x + sin 3x (cos 3x. sin 3 x + sin 3x. cos 3 x) = sin x. sin 2 3x

3 sin 4 x

95. (1 + sin x) 2 = cos x
3
2

96. tan 2 x + sin 2 x = cot x

97. sin 2 x + 2 cos 2 x = 1 + sin x − 4 cos x
98. sin 2 x + 2 tan x = 3
99. 3 - tanx ( tanx + 2sinx) + 6cosx = 0
100. 4 cos 2 x + 3 tan 2 x − 4 3 cos x + 2 3 tan x + 4 = 0
101. cos 2 3x. cos 2 x − cos 2 x = 0
102. 1 + sin x + cos 3 x = cos x + sin 2 x + cos 2 x
103. sin 2 x(tan x + 1) = 3 sin x(cos x − sin x) + 3


2 − 3 ) cos x − 2sin
28. (

2

2 cos x − 1

104. sin 3 x + sin 2 x = 5 sin x
105.T×m c¸c nghiÖm thuéc [ 2;40] cña ph¬ng tr×nh sau:

x π
 − ÷
 2 4  =1

sin x − cos 2 x = 0

cos x ( cos x − 1)
= 2 ( 1 + sin x )
sin x + cos x
2 cos 4 x
30. cot x = tan x +

sin 2 x

cos 2 x sin 2 x
+
)
sin x
cos x
107. cos 2 x + (1 + 2 cos x)(sin x − cos x) = 0
π
108. 2 sin( 2 x − ) + 4 sin x + 1 = 0
6

2

106. 3 + cot 2 = 3(

29.

31. tan
32.
33.

4

( 2 − sin
x +1 =

2

2 x ) sin 3 x

2 sin 2 x + 2 cos x − 2 sin x − 1
= cos 2 x + 3 (sin x + 1)
2 cos x − 1
110. sin 3 x + cos 3 x = 3 (1 + sin 2 x )(cos x − sin x )
2

109.

cos 4 x

sin 4 x + cos 4 x 1
1
= cot 2 x −
5sin 2 x
2
8sin 2 x.

111. 2 cos 3 x(2 cos 2 x + 1) = 1

1
= sin x
8cos 2 x

2
112. cos x(cos x − 1) = 2(1 + s inx)

sin x
 3π


− x ÷+
=2
34. tan 
 2
 1 + cos x
35. sin7x + sin5x + sin3x = 0
36. sinx+sin2x+sin3x+sin4x=0
37. cos2x- cos8x+ cos4x=1
38. 2sinx.cos2x – 1 + 2cos2x – sinx = 0
39. sin2 x + sin2 3x = cos2 2x + cos2 4x
40. sinx+2cosx+cos2x-2sinxcosx=0
41. sin2x-cos2x=3sinx+cosx-2
42. sin3 x+2cosx-2+sin2 x=0
43. 3sinx+2cosx=2+3tanx
44. 2sin2x-cos2x =7sinx+2cosx-4
45. 1+ sinx+ cos3x= cosx+ sin2x+ cos2x
46. 1+sinx+cosx+sin2x+cos2x=0
47. cos3x+cos2x+2sinx-2=0
48. cos2x-2cos3x+sinx=0
49. sin2x=1+ 2 cosx+cos2x
π

x

50. sin x. cos 4 x + 2 sin 2 2 x = 1 − 4 sin 2 ( − )
4 2
51. 1+tanx=sinx+cosx
52. (1-tanx)(1+sin2x)=1+tanx
53. cotx-tanx=cosx+sinx
54. 9sinx+6cosx-3sin2x+cos2x=8

55. sin8x+ cos4x=1+2sin2xcos6x
56. cosx+cos2x+cos3x+cos4x=0
57. 3cosx+cos2x- cos3x+1=2sinxsin2x
58. sin5x+ sinx+2sin2x=1
59. 3 + 2sinx.sin3x = 3cos2x
60. 2cosx.cos2x = 1 + cosx + cos3x
61. sin x cos 2 x + cos 2 x (tan 2 x − 1) + 2sin 3 x = 0

s inx + cos x

113. 2 sin 2 x − sin 2 x + s inx + cos x − 1 = 0
114. 3 sin 2 x(2 cos x + 1) + 2 = cos3 x + cos2 x − 3cos x
115. 2sin 3 x − cos 2 x + cos x = 0
116. cos8 x + 3cos 4 x + 3cos 2 x = 8cos x.cos3 3 x − 1
117.

2

3 cos 5x − 2sin 3x cos 2x − sin x = 0
π
π 1



118. cos  x − ÷+ cos  x + ÷ = cos 2 x − 1
4
4
3






119. cos22x – 2cos(x +



3π
π
) sin(3 x − ) = 2
4
4

120. 4(sin 4 x + cos 4 x) + 3 sin 4 x = 2
121. sin5x+ sinx+2sin2x=1
122. Cos2x + 3sin2x +5Sinx – 3Cosx =3
123. cos2x + cos4x + cos6x = cosxcos2xcos3x + 2
π

124. cos x + cos3x = 1 + 2 sin  2x + ÷

4
3
2 sin x
125. ( 2 cos x − 1) cot x =
+
s inx cos x − 1


126.

sin 4 2 x + cos 4 2 x
tan(

π

4

− x).tan(

π

4

+ x)

= cos 4 4 x

127. cos3x + sin7x = 2sin2(

π 5x
9x
) - 2cos2
.
+
4 2
2

128. (1 + cos 2 x) sin 2 x = 2(sin 3 x + sin x )(1 + sin x)

1 − sin x
π

2sin  − 2 x ÷ + 3 cos 4 x = 3 − 4sin 2 x.
4



sin x
 3π

=2
62. tan  − x ÷+
2
1
+
cos x



129.

63. sin 2 x + cos 2 x + 3sin x − cos x − 2 = 0
64. 4(sin 3 x + cos3 x) = cos x + 3sin x

130. 4 cot x − 2 =

2

3 + cos 2 x

sin x

π
131. sin x.sin 2 x − cos x.sin 2 2 x + 2sin 2 ( − x) = 1

132.

4
3(sin 2 x − 3sin x) + 3 = 2 cos x + 3cos x − 2
2

π
4

133. sin 2 x + (1 + 2 cos 3 x) s inx-2sin 2 (2 x + ) = 0
ph¬ng tr×nh lîng gi¸c (Trong c¸c ®Ò thi ®¹i häc vµ cao ®¼ng Tõ 2002 ®Õn 2010)


1.(KA-2009) Gi¶i ph¬ng tr×nh:

(1 − 2 sin x). cos x
= 3
(1 + 2 sin x)(1 − sin x)

2.(KB-2009) Gi¶i ph¬ng tr×nh: sin x + cos x. sin 2 x + 3 cos 3 x = 2(cos 4 x + sin 3 x)
3.(KD-2009) Gi¶i ph¬ng tr×nh: 3 cos 5 x − 2 sin 3 x. cos 2 x − sin x = 0
4.(C§-2009) Gi¶i ph¬ng tr×nh: (1 + 2 sin x) 2 cos x = 1 + sin x + cos x

π
1

1
)+
=
4
cos x sin x
6
6
6. C§ KT §èI NGO¹I 2007 Gi¶i ph¬ng tr×nh: sin x + cos x = 2 sin 2 x
5. C§ ytÕ HN n¨m 2007 Gi¶i ph¬ng tr×nh: 2 2 sin( x −

7. C§ 2008: Gi¶i ph¬ng tr×nh: sin 3x − 3 cos 3x = 2 sin 2 x
8. KD - 2008: Gi¶i ph¬ng tr×nh: 2 sin x(1 + cos 2 x) + sin 2 x = 1 + 2 cos x

1
1
7π
+
= 4 sin(
− x)
9. KA - 2008: Gi¶i ph¬ng tr×nh: sin x
3π
4
sin( x − )
2
10. KB - 2008: Gi¶i ph¬ng tr×nh: sin 3 x − 3 cos 3 x = sin x cos 2 x − 3 sin 2 x cos x
x
x
11. KD - 2007: Gi¶i ph¬ng tr×nh: (sin + cos ) 2 + 3 cos x = 2
2
2

12. KB - 2007: Gi¶i ph¬ng tr×nh: 2 sin 2 2 x + sin 7 x − 1 = sin x
13. KA - 2007: Gi¶i ph¬ng tr×nh: (1 + sin 2 x) cos x + (1 + cos 2 x) sin x = 1 + sin 2 x
14. KD - 2006: Gi¶i ph¬ng tr×nh: cos 3x + cos 2 x − cos x − 1 = 0
x
15. KB - 2006: Gi¶i ph¬ng tr×nh: cot x + sin x (1 + tan x tan ) = 4
2

2(cos 6 x + sin 6 x) − sin x cos x

=0
2 − 2 sin x
17. KB - 2005: Gi¶i ph¬ng tr×nh: 1 + sin x + cos x + sin 2 x + cos 2 x = 0
16. KA - 2006: Gi¶i ph¬ng tr×nh:

18. KA - 2005: Gi¶i ph¬ng tr×nh: cos 2 3 x.cos 2 x − cos 2 x = 0
19. KD - 2004: Gi¶i ph¬ng tr×nh: ( 2 cos x − 1)(2 sin x + cos x ) = sin 2 x − sin x
20. KB - 2004: Gi¶i ph¬ng tr×nh: 5 sin x − 2 = 3(1 − sin x) tan 2 x

x
2

21. KD - 2003: Gi¶i ph¬ng tr×nh: sin 2 ( −

π
x
) tan 2 x = cos 2
4
2

2

sin 2 x
cos 2 x
1
23. KA - 2003: Gi¶i ph¬ng tr×nh: cot x − 1 =
+ sin 2 x − sin 2 x
1 + tan x
2
24. KD - 2002: Gi¶i ph¬ng tr×nh: cos 3 x − 4 cos 2 x + 3 cos x − 4 = 0
25. KB - 2002: Gi¶i ph¬ng tr×nh: sin 2 3 x − cos 2 4 x = sin 2 5 x − cos 2 6 x
cos 3 x + sin 3x
26. KA - 2002: Gi¶i ph¬ng tr×nh: 5(sin x +
= cos 2 x + 3
1 + 2 sin x
22. KB - 2003: Gi¶i ph¬ng tr×nh: cot x − tan x + 4 sin 2 x =

27. KD - 2010 : Giải phương trình : sin 2 x − cos 2 x + 3sin x − cos x − 1 = 0
28. KB - 2010 : Giải phương trình : (sin 2x + cos 2x) cosx + 2cos2x – sin x = 0
π

(1 + sin x + cos 2x)sin  x + ÷
1
4
29. KA - 2010: Giải phương trình

=
cos x
1 + tan x
2

D¹ng 1: Ph¬ng tr×nh bËc nhÊt ®èi víi 1 hµm sè lîng gi¸c.
1. sin 2 3 x − cos 2 4 x = sin 2 5 x − cos 2 6 x
2. sin x + sin 2 x + sin 3 x = 0
3. sin x + cos 2 x − cos 4 x = 0
1
4. cos x cos 2 x cos 4 x cos8 x =
16
1 − cos 4 x
sin 4 x
5.
=
2sin 2 x 1 + cos 4 x
sin x.cot 5 x
6.
=1
cos 9 x
7. 4 3 sin x cos x cos 2 x = sin 8 x
8. 1 + tan x = 2 2 sin x
1
2(cos x − sin x)
9.
=
tan x + cot 2 x
cot x − 1
2
10. 2 tan x + cot x = 3 +
sin 2 x
11. tan x + cot x = 2(sin 2 x + cos 2 x)
12. cos3 x − sin 3 x = sin x − cos x
13. sin x + sin 2 x + sin 3 x = cos x + cos 2 x + cos 3 x

14. 2 cos3 x + cos 2 x + sin x = 0
15. (2sin x + 1)(3cos 4 x + 2sin x − 4) + 4 cos 2 x = 3
16. sin 6 x + cos 6 x = cos 4 x
5
17. sin 8 x + cos8 x = 2(sin10 x + cos10 x) + cos 2 x
4
18. 3 − cos x − cos x + 1 = 2
D¹ng 2: Ph¬ng tr×nh bËc hai ®èi víi 1 hµm sè lîng gi¸c.
19. 3cos x + cos 2 x − cos 3 x + 1 = 2sin x sin 2 x
1
1
2
20.
+
=
cos x sin 2 x sin 4 x
21. cos x(cos x + 2sin x) + 3sin x(sin x + 2) = 1
sin 2 x − 1
7
π
π
22. sin 4 x + cos 4 x = cot( x + ) cot( − x)
8
3
6
23. 4sin 3 x + 3cos 3 x − 3sin x − sin 2 x cos x = 0
24. (1 − cos x)

+ (1 + cos x) 2
1 + sin x

− tan 2 x sin x =
+ tan 2 x
4(1 − sin x)
2
2

25. 2sin 2 x(4sin 4 x − 1) = cos 2 x(7 cos 2 2 x + 3cos 2 x − 4)
26. 3cos 4 x − 4 cos 2 x sin 2 x + sin 4 x = 0

π
27. sin 3 ( x − ) = 2 sin x
4
28. 4sin 2 2 x + 6sin 2 x − 9 − 3cos 2 x = 0
29. cos 4 x − 5sin 4 x = 1
30. 4sin 2 x + 3 tan 2 x = 1
1
31. 2 tan x + cot 2 x = 2sin 2 x +
sin 2 x
32. sin 3 x + sin 2 x = 5sin x
sin 3 x sin 5 x
33.
=
3
5
34. sin 4 x = tan x
35. cos x cos 4 x + cos 2 x cos 3 x = 0
1
1
36. 2sin 3 x −
= 2 cos 3 x +

sin x
cos x
37. 5sin x + cos 2 x + 2 cos x = 0
38. 5cos x − cos 2 x + 2sin x = 0
1
39. ( 1 − cos x + cos x ) cos 2 x = sin 4 x
2
D¹ng 3: Ph¬ng tr×nh bËc ba ®èi víi 1 hµm sè lîng gi¸c.
π
40. (sin 2 x + 3 cos 2 x) 2 − 5 = cos(2 x − )
6
π
41. 8cos3 ( x + ) = cos 3 x
3
42. tan x = cot x + 2 cot 3 2 x
43. 4 cos3 2 x + 6sin 2 x = 3
44. cos x + cos 2 x + cos 3 x + 1 = 0
45. sin 2 x = cos 2 2 x + cos 2 3 x
46. cos 3 x − 2 cos 2 x = 2
13
47. cos6 x − sin 6 x = cos 2 2 x
8
48. 3 tan 2 x − 4 tan 3 x = tan 2 3 x tan 2 x
49. cos3 x − 4sin 3 x − 3cos x sin 2 x + sin x = 0
50. 1 + 3 tan x = 2sin 2 x
51. sin 3 x − 2sin x = 4 cos3 x − 3
1
52. 2 cos 2 x − 8cos x + 7 =
cos x
53. sin x sin 2 x + sin 3 x = 6 cos 3 x

54. tan x − tan 2 x = sin x
55. 4 cos x − 2 cos 2 x − cos 4 x = 1
56. 4 cos3 x − cos 2 x − 4 cos x + 1 = 0
57. cos 3 x − 2 cos 2 x + cos x = 0


Dạng 4: Phơng trình bậc 4 trùng phơng đối với 1
hàm số lợng giác.
sin 5 x
58.
=1
5sin x
59. 1 sin 2 x + 1 + sin 2 x = 4 cos x
sin x
1
cos5 x + sin 7 x + (cos3 x + sin 5 x)sin 2 x
60.
2
= cos x + sin x
61. sin 6 x + cos 6 x = 2(sin 8 x + cos8 x)
1
62. sin 8 x + cos8 x + cos 4 x = 0
8
3
63. tan 2 x + sin 2 x = cot x
2
64. cot x = tan x + 2 tan 2 x

65. tan 3 ( x ) = tan x 1
4

3
66. cos 2 x + cos 2 2 x + cos 2 3x + cos 2 4 x =
2
4
4
67. sin x cos x = 2 3 sin x cos x + 1
68. 2 2(sin x + cos x) cos x = 3 + cos 2 x
69. 4sin 2 x 3cos 2 x = 3(4sin x 1)
70. cos3 x + sin 3 x = sin 2 x + sin x + cos x

1
71. cos 4 x + sin 4 ( x + ) =
4
4
72. 9sin x + 6 cos x 3sin 2 x + cos 2 x = 8
73. tan x 3cot x = 4(sin x + 3 cos x)
74. sin x 4sin 3 x + cos x = 0
cos3 x sin 3 x
75.
= 2 cos 2 x
sin x + cos x
76. sin x cos x + 2sin x + 2 cos x = 2


77. sin 3 x + cos3 x = 1 tan( + x).tan( x)
4
4
1
78. cos3 x + sin 3 x = 1 sin 2 x
2

79. sin 2 x(sin x + cos x) = 2
80. 2sin 3 x cos 2 x + cos x = 0
81. sin 3 x + cos3 x = 2sin 2 x + sin x + cos x
82. 2sin 3 x sin x = 2 cos 3 x cos x + cos 2 x
3
83. tan 2 x = 1 cos3 x
1 sin x
2
84. cos x + sin 3 x + cos x = 0
3(1 + sin x)
x
85. 3 tan 3 x tan x +
8cos 2 ( )
2
cos x
4 2
1. sin 2 x cos 2 x + 3sin x cos x 1 = 0

86. 2sin x + cot x = 2 sin 2 x + 1
87. 3(cot x cos x) 5(tan x sin x) = 2


88. sin 2 x + 2 sin x ữ = 1
4

89. sin 2 x + 4(cos x sin x) = 4
Dạng 5:Phơng trình đẳng cấp đối với sinx và
cosx.
90. 1 + 3 sin 2 x = ( 3 1) cos 2 x + 1
2

91.
2sin 2 2 x 10sin 2 x + cos 4 x + 40 cos 2 x + 1 = 0
92. 3sin 3 x sin 2 x.cos x + 4 cos3 x = 0
1
93. 4sin x + 6 cos x =
cos x
Dạng 6:Phơng trình lợng giác không mẫu mực.
8x
10 x
94. cos cos 2 x cos
= 2sin 2 x + 1
3
3
95. sin 8 x + cos 2 x sin 2 x = 2 cos 2 x + 1
96. 4 cos x 2 cos 2 x cos 4 x = 1
97. sin 3 x(cos x 2sin 3 x) + cos 3 x(1 + sin x 2cos 3 x) = 0
x
x
98. 4 cos3 3sin 2 x cos 2 x = 3cos
4
4
99. sin 3 x + cos 4 x 4sin 7 x = cos10 x + sin17 x
100. 8cos 4 x.cos 2 2 x + 1 cos 3 x + 1 = 0
101. sin x + sin x + sin 2 x cos x = 1
102.
sin 2 3x
2
sin x +
(cos 3 x.sin 3 x + sin 3 x.cos 3 x)
3sin 4 x

= sin x.sin 2 3 x
103. 2sin 2 x cos 2 3x = cos 2 5 x cos 2 x 1

104. 2sin 2 ( x + ) + 1 sin 6 x = 0
3
105. cos 2 x cos 6 x + 4(3sin x 4sin 3 x + 1) = 0
106. sin 2 x + 2 = sin 2 3 x + 2sin x
107. |sin x | =| cos x |
108. cos 3x + 2 cos 2 3x = 2(1 + sin 2 2 x)
109. sin x + cos x = 2(2 sin 2 x)
110. sin 2009 x + cos 2009 x = 1
111. cos13 x + sin14 x = 1
112. 2|sin x| + | sin x |= cos x cos 2 x + 1
113. (cos 4 x cos 2 x) 2 = 5 + sin 3 x


2. 9 sin x + 6 cos x − 3sin 2 x + cos 2 x = 8
3.