Phương trình lượng giác và ứng dụng nâng cao

)

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

 

22
sin x cos x 1

tanx.cotx 1

sin x
tan x
cos x


cos x
cotx
sin x

os
2
2
1
1 tan x
cx

2
2
1

1 cot x
sin x

     

sin2x 2sinx.cos x


22
22
cos x sin x
cos2x
2cos x 1 1 2sin x


os
2
1 c 2x
sin x
2

os
os
2
1 c 2x
cx
2


3

sin3x 3sinx 4sin x

3
cos3x 4 cos x 3cosx

 

sin a b sina.cosb cosa.sinb

osc a b cosa.cosb sina.sin b


tana tan b
tan a b
1 tana.tan b

tana tan b
tan a b
1 tana.tan b


π 1 tan x
tan x
4 1 tan x

π 1 tan x
tan x
4 1 tan x

 


a b a b
cosa cosb 2 cos .cos
22

a b a b
cosa cos b 2sin .sin
22


a b a b
sina sin b 2sin .cos
22

a b a b
sina sin b 2cos .sin
22


sin a b
tana tan b
cosa.cos b

sin a b
tana tan b
cosa.cos b

 

cos a b cos a b

cosa.cosb
2

sin a b sin a b
sina.cosb
2


cos a b cos a b
sina.sin b
2

 

ππ
sin x cos x 2 sin x 2 cos x
44

ππ
sin x cos x 2 sin x 2 cos x
44


4 4 2
1 cos 4x
cos x sin x 1 s
31
in 2x
2 4


6 6 2
3 cos 4x
cos x sin x 1 s
53
in 2x
4 8


Ths.   (Nâng cao)


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 :
 
sin x
cos x

11
.
 
tan

cot


 
tanx

cos x 0 x k k
2

.
 
cotx

sin x 0 x k k
.
 
tanx

cotx

x k. k
2
.
 

 
x


 

"
AM


k2
n

0
0
k.360
hay a
n

k ,n

n

M

".

AM k2
3

M

3

k0
).

AM k
6

M

6


7
6

k 0,k 1
).

2
AM k.
43

M

11
;
4 12

19
12
,
k 0;1;2
.

k2
AM k.
4 2 4 4

M

4

,
3
4
,
5
4
;
7
4


k 0,1,2,3
).

xk
6

xk
3


xk
6

6

5
6



xk
3
 
3

4
3
.


xk
32

 
2
2
11
cos x cos x
22
11
sin x sin x
22




/3
5/6
4/3
/6

O
)

Page - 3 -
2
2
2
2
1
cos x
2 cos x 1 0 cos2x 0
2
1 cos2x 0
2 sin x 1 0
sin x
2

2
2
sin x 1 sin x 1
cos x 1
cos x 1

22
sin x cos x 1

22
22
sin x 1 cos x 0 cos x 0
sin x 0

cos x 1 sin x 0

 '' c s p''
 ''''

 
cos cos

''  ''
sin sin , tan tan , cot tan

 
sin sin

''  ''
cos cos , tan tan , cot tan

 
0


sin cos , cos sin , tan cot , cot tan
2 2 2 2

  '' '' 

sinu cosv


sinu sinv


sinu cosv


sin u cos v sin u sin v
2


u v k2 u v k2 , k
22
.

2
sin x cos x
3


 

sin x k2 sin x
cos x k2 cos x


sin x k2 sin x
k
cos x k2 cos x
.

Ths.   (Nâng cao)


Page - 4 -

 :
u v k2
sin u sin v
u v k2
:
sin x 0 x k
sin x 1 x k2
2
sin x 1 x k2
2

 :
u v k2
cos u cos v
u v k2
:
cos x 0 x k
2
cos x 1 x k2
cos x 1 x k2

 :
tan u tan v u v k
Ðk : u,v k

2
:
tan x 0 x k
tan x 1 x k
4

 :
cotu cotv u v k
Ðk : u,v k
:
cotx 0 x k
2
cotx 1 x k
4




cos3x 4cos2x 3cos x 4 0 , x 0;14


2cos x 1 2sin x cos x sin2x sin x


cos3x cos2x cosx 1 0


sin x cos x 1 sin2x cos2x 0



sin x 1 cos2x sin2x 1 cos x



1 1 7
4 sin x
sin x 4
3
sin x
2



44
7
sin x cos x cot x cot x
8 3 6



44
4
sin 2x cos 2x
cos 4x
tan x tan x
44

9

3 x 1 3x

sin sin 1
10 2 2 10 2


sin 3x sin2x sin x 1
44

)

Page - 5 -
.

3
8 cos x cos 3x 1
3



3
2 sin x 2 sin x 1
4



3
sin x 2 sin x 1
4


cosx cos2x cos3x cos4x 0



2 2 2
3
sin x sin 2x sin 3x
2
.


2 2 2
sin x sin 2x sin 3x 2
.


2 2 2 2
sin x sin 3x cos 2x cos 4x



2 2 2 2
sin 3x cos 4x sin 5x cos 6x


sin
22
5x 9x
cos 3x sin7x 2 2cos
4 2 2



2 2 2
sin x cos 2x cos 3x



2
2sin 2x sin7x 1 sin x

. 
sin x sin2x sin 3x 1 cos x cos2x



3 3 3
sin x cos3x cos x sin3x sin 4x



23
cos10x 2cos 4x 6cos3x cos x cos x 8 cos x cos 3x



3 3 2
4sin x 3cos x 3sin x sin x cosx 0

nh:

2
2sin x 1 3cos4x 2sin x 4 4cos x 3



6 6 8 8
sin x cos x 2 sin x cos x



8 8 10 10
5
sin x cos x 2 sin x cos x cos2x
4



3 3 5 5
sin x cos x 2 sin x cos x



4 2 2 4
3cos x 4cos x sin x sin x 0



33
2 3 2
cos 3x cos x sin 3x sin x

8



1
cos x cos2x cos4x cos8x
16



3
4sin 3x cos2x 1 6sin x 8sin x



1
cos x cos2x cos3x cos4x cos5x
2



sin2x 2 cos x sin x 1
0
tan x 3

Ths.   (Nâng cao)


Page - 6 -


2
1 sin2x cos2x
2 sin x sin2x
1 cot x


tan x cotx 2 sin2x cos2x



2
tan x tan x tan3x 2



222
11
tan x cot x cot 2x
3



2 2 2
xx
sin tan x cos 0
2 4 2



2
sin2x cotx tan2x 4 cos x



22
cot x tan x
16 1 cos4x
cos2 x



1
2 tan x cot2x 2 sin2x
2sin2x



3 sin x tan x
2 1 cos x 0
tan x sin x



22
22
1 cos x 1 cos x
1
tan x sin x 1 sin x tan x
2

4 1 sin x


cos3x tan5x sin7x



11
sin2x sin x 2cotx
2sin x sin2x



44
sin x cos x 1
tan x cot2x
sin2x 2



2 2 2 2
tan x.cot 2x.cot3x tan x cot 2x cot3x



x
cotx sin x 1 tan x tan 4
2


Câu 1. 
2sin x cosx 2cos x 3 3 sin x
.
Câu 2. 
2tanx cosx 1 2cosx tanx
.
Câu 3. nh:
33
2
sin x cos x cos x sin x
8
.
Câu 4. 
2 2 2
cos x cos 2x cos 3x 1
.
Câu 5. 
22
17
sin 2x cos 8x sin 10x
2
.
Câu 6. 
46
cos x sin x cos2x
.
)

Page - 7 -

Câu 7. 
1 cos 4x sin 4x
0
2sin2x 1 cos4x
.
Câu 8. 
2
21
sin x cos x cos x
2
.
Câu 9. 
2
x
2 3 cos x 2sin
24
1
2cos x 1
.
Câu 10. 
sin4x 3sin2x tanx
.
Câu 11. 
23
cos10x 2cos 4x 6cos3x cosx cosx 8cosx cos 3x
.
Câu 12. 
2 2 2
2cos x 2cos 2x 2cos 3x 3 cos4x 2sin2x 1
.

Câu 13. 

5x 7
sin 2x 3 cos x 1 2sin x , ;3
2 2 3
.
Câu 14. 

22
sin 4x cos 6x sin 10,5 10x , 0;
2
.
Câu 15. 
tan2x tan3x tan5x tan2x tan3xtan5x
.
Câu 16. 
sin x sin2x sin 3x
3
cos x cos2x cos 3x
.
Câu 17. 
2
1 cos x
tan x
1 sin x
.
Câu 18. 
2
4
cos x cos x

3
.
Câu 19. 
11
2 2 sin x
4 sin x cos x
.
Câu 20. 
2
2 tan x cot2x 3
sin2x
.
Câu 21. 
2
3 tan 3x cot2x 2 tan x
sin4 x
.
Câu 22. 
2 2 2
sin x sin 2x sin 3x 2
.
Câu 23. 
25 4x 3 sin2 x 8sin x 0
.
Câu 24. 
sin2x
2cos x 0
1 sin2x
.
Câu 25. 

sin x cot5x
1
cos9x
.
Ths.   (Nâng cao)


Page - 8 -
Câu 26. 
2
3 tan6x 2 tan2x cot4x
sin8 x
.
Câu 27. 
2
1 cos x
tan x
1 sin x
.
Câu 28. 
33
2
cos x cos 3x sin x sin 3 x
4
.
Câu 29. 
44
x x 5
sin cos
3 3 8

.
Câu 30. 
2
2sin 3x 1 4 sin x 1
.
Câu 31. 
3 3 2
cos x 4sin x 3cosx sin x sinx 0
.
Câu 32. 
44
xx
sin cos 1 2 sin x
22
.
Câu 33. 
sin 3x sin2x sin x
44
.
Câu 34. 
2
4
4
2 sin x sin3x
tan x 1
cos x
.
Câu 35. 
2
x

tan x cos x cos x sin x 1 tan tan x
2
.
Câu 36. 

22
x7
sin x cos4x 2 sin 2x 4 sin x , x 1 3
4 2 2
.
Câu 37. 
sinx sin2x sin3x 1 cosx cos2x
.
Câu 38. 
2 2 2 2
cos x cos 2x cos 3x cos 4x 2
.
Câu 39. 
2
cos x cos x 1
2 1 sin x
sin x cos x
.
Câu 40. 
sinx sin2x sin3x sin4x sin5x sin6x 0
.
Câu 41. 
cosx cos3x 2cos5x 0
.
Câu 42. 

9sinx 6cos x 3sin2x cos2x 8
.
Câu 43. 
cosx sin x cos x sin x cos x cos2x
.
Câu 44. 
2
2sin x 1 3cos4x 2sin x 4 4cos x 3
.
Câu 45. 
33
2sin x sinx 2cos x cosx cos2x
.
Câu 46. 
2 3 4 2 3 4
sinx sin x sin x sin x cosx cos x cos x cos x
.
Câu 47. 
2
sin x tan x 1 3sin x cosx sin x 3
.
Câu 48. 
2
tan2x cotx 8cos x
.
Câu 49. 
3 cotx cos x 5 tan x sin x 2
.
Câu 50. 
11

2 2 sin x
4 sin x cos x
.
)

Page - 9 -










2
a sin x bsin x c 0

t sinx

1 t 1

2
a cos x bcosx c 0

t cosx

1 t 1

2
a tan x btanx c 0


t tanx

x k , (k )
2

2
a cot x bcotx c 0


t cotx

x k , k

t
2
t sin x
hoc
t sin x
u kin là
0 t 1


cos
)



2
22
1 sin2x sin x cos x 2sin x cos x sin x cosx


2
22
1 sin2x sin x cos x 2sin x cos x sin x cos x


sin2x
sin x cos x
2


33
sin x cos x sin x cos x 1 sin x cos x


x
33
sin x cos x sin x cos x 1 sinx cos


22
sin x cosx sin x cos x 2
tan x cotx

cos x sin x sin x cos x sin2x


22
cos x sin x cos x sin x 2cos2x
cotx tan x 2cotx
sin x cos x sin x cos x sin2x


4 4 2 2
1 1 1 3 1cos 4x
sin x cos x 1 sin 2x cos 2x
2 2 2 4


4 4 2 2 2 2
cos x sin x sin x cos x cos x sin x cos2x


6 6 4 4 2 2 2
3 5 3cos4x
sin x cos x sin x cos x sin x cos x 1 sin 2x
48


6 6 4 4 2 2
cos x sin x cos2x sin x cos x sin x cos x


x1

1 tan x tan
2 cos x


sin x cos x 2 sin x 2 cos x
44


22
cos x cos x 1 sin x 1 sin x
1 sin x cos x
cos x 1 sin x cos x 1 sin x


Ths.   (Nâng cao)


Page - 10 -



2
cos4x 12sin x 1 0

. 

44
cos x sin x cos4x 0



cos 3x sin3x
5 sin x 3 cos2x , x 0;2
1 2sin2x



sin 3x sin5x
35



sin5x
1
5sin x



22
cos 3xcos2x cos x 0 1



44
3
cos x sin x cos x sin 3x 0
4 4 2



57
sin 2x 3 cos x 1 2 sin x; x ;2
2 2 2



2
5sin x 2 3 1 sin x tan x



2
sin2x 3 cos2x 5 cos 2x
6



66
2 cos x sin x sin x cos x
0
2 2 sin x



1 sin x cos2x sin x
4
1
cos x
1 tan x
2



11
2sin 3x 2cos 3x
sin x cos x



2
cos x 2sin x 3 2 2cos x 1
1
1 sin2x



x 3x x 3x 1
cos x cos cos sin x sin sin
2 2 2 2 2



44
sin x cos x 1 1
cot2x
5sin2x 2 8sin2x



22

3cot x 2 2 sin x 2 3 2 cos x



62
3cos4x 8cos x 2cos x 3 0



2cos 4x
cotx tan x
sin2x



2
cotx tan x 4 sin2x
sin2x



22
2sin x tan x 2

)

Page - 11 -


88

1
sin x cos x
8



8 8 2
17
sin x cos x cos 2x
16



3
5x x
sin 5cos x sin
22



2
sin2x cotx tan2x 4 cos x



2
6x 8x
2cos 1 3cos
55



3
tan x tan x 1
4



44
4
sin 2x cos 2x
cos 4x
tan x tan x
44



42
12
48 1 cot2x cotx 0
cos x sin x



8 8 10 10
5
sin x cos x 2 sin x cos x cos2x
4

.

2
cos2x 1
cotx 1 sin x sin2x
1 tan x 2


sin2x 2tan x 3


1 tan x 1 sin2x 1 tan x



2
cos2x cosx 2 tan x 1 2


3
sin2x cos x 3 2 3 cos x 3 3 cos2x 8 3 cos x sin x 3 3


2
2
11
4 sin x 4 sin x 7 0
sin x
sin x



2
tan x tan x tan3x 2



11
sin2x sin x 2cot2x
2sin x sin2x

9

1
2cos2x 8 cos x 7
cos x


4 sin3x cos2x 5 sin x 1


2
2
11
sin x sin x
sin x
sin x




2
2
11
cos x 2 cos x 1
cos x
cos x



1
2 tan x cotx 2sin2x
sin2x



22
tan x cot x 2 1 tan x cotx 0


Ths.   (Nâng cao)


Page - 12 -


44
11
sin x cos x sin2x
8


5 5 2
4sin x cosx 4cos xsinx sin 4 x



cos 2x cos 2x 4 sin x 2 2 1 sin x
44


2 2 2 2
3
cos x cos 2x cos 3x cos 4x
2


.
66
22
sin x cos x 1
tan2x
4
cos x sin x



66
sin x cos x sin2x


Câu 51. 
3
4cos x 3 2 sin2x 8 cos x

Câu 52. 
2
6sin 3x cos12x 14

Câu 53. 
3 tan x cotx 1 3

Câu 54. 
tan x 3 cotx 1 3

Câu 55. 
22
13
4
sin x cos x
sin x cos x

Câu 56. 
2
1
4 tan x 2 0
cos x

Câu 57. 

2
1
cotx 3
sin x

Câu 58. 
2
22
1 2 2 sin x
1 cot x

Câu 59. 
2
4
cos x 9 0
1 tan x

Câu 60. 
44
17
2 sin x cos x cos 2x 0
2

Câu 61. 
sin4x tanx

Câu 62. 
4 4 4
9
sin x sin x sin x

4 4 8

Câu 63. 
tanx cotx 4

Câu 64. 
2
sin x 3 2 2cos x 2sin x 1
1
1 sin2x

Câu 65. 
4
4cos x 3 2 sin2x 8cosx

Câu 66. 
1 1 2
cos x sin2x sin 4x

Câu 67. 
sin2x 2 sin x 1
4

)

Page - 13 -
Câu 68. :
2 2sin x 1 4 sin x 1 cos 2x sin 2x
44

Câu 69. 
2
4x
cos cos x
3

Câu 70. 
x
tan cos x sin2x 0
2

Câu 71. 
1 3tanx 2sin2x

Câu 72. 
2
3x 4x
2cos 1 3cos
55

Câu 73. 
cotx tanx 2tan2x

Câu 74. 
2
3x
2cos 1 3 cos2x
2

Câu 75. 

2
3cos4x 2cos 3x 1

Câu 76. 
x
cos x tan 1
2

Câu 77. 
2
3tan2x 4 tan3x tan 3x tan2x

Câu 78. 
2
3
cos x cos4x cos2x cos3x cos 4x
2

Câu 79. 
1 tan x 1 sin2x 1 tanx

Câu 80. 
6 6 2
13
sin x cos x cos 2x
8

Câu 81. 
66
1

sin x cos x cos2x
16

Câu 82. 
2
5sin x 2 3 tan x 1 sin x

Câu 83. 
4
4
1
sin x sin x 1
8

Câu 84. 
2
8x 2x
cos cos
33

Câu 85. 
2
x
cos2 x 3 cos x 4cos
2

Câu 86. 
2
cos5x cosx cos4x cos2x 4 3sin x

Câu 87. 
2cosxcos2x 1 cos2x cos3x

Câu 88. 
sin3x cos2x 2 sin2x cos x 1

Câu 89. 
44
2 cos 2x sin 2x cos8x cos4x 0

Câu 90. 
2
x 5 17 1 9
2cos 2 2x cos 10cos x sin x
2 2 2 2 2

Câu 91. 
5
4 cos x cos 2x
3 3 2

Câu 92. 
44
5
sin x cos x 2sin x
2

Câu 93. 
66
1

sin x cos x sin2x
4

Ths.   (Nâng cao)


Page - 14 -
Câu 94. 
22
3sin 2x 8sin x 11 3cos2x
0
sin2x

Câu 95. 
2
cos2x 3 2 2 sin x 3 2sin x
10
sin2x 1

Câu 96. 
sin x 1
1
1 cos2x

Câu 97. 
cos2x 3 cot2x sin4x
2
cot2x cos2x

Câu 98. 

44
4
xx
sin cos
22
cos x
tan x tan x
44

Câu 99. 
2
2
11
cos x cos x
cos x
cos x

Câu 100. 
cos2x 5 2 2 cosx sin x cosx

)

Page - 15 -


:
a sin x bcos x c , a,b \ 0

 :
 

2 2 2
a b c

 Chia 2
22
a b 0

2 2 2 2 2 2
a b c
sin x cos x
a b a b a b
.
 

2 2 2 2
ab
sin ;cos , 0;2
a b a b

2 2 2 2
cc
sin sin x cos cos x cos(x )
a b a b

 :
 
xx
cos 0 k x k2
2 2 2



 
xx
cos 0 k x k2
2 2 2



2
22
x 2t 1 t
t tan sin x , cos x
2
1 t 1 t


2
(b c)t 2at c b 0
.
 
x k2 b c 0
nên 
2 2 2 2 2 2
' a (c b ) 0 a b c
.
  
t

x
tan t x

2




. 

26
cos7x 3 sin7x 2 , x ;
57



2
xx
sin cos 3 cos x 2
22



1
tan x sin2x cos2 x 2 2cos x 0
cos x



1 2 sin x cos x
3
1 2 sin x 1 sin x

Ths.   (Nâng cao)


Page - 16 -


31
8 sin x
cos x sin x



3
sin x cosx sin2x 3 cos3x 2 cos4x sin x



3
3sin3x 3 cos9x 1 4 sin 3x


3 cos5x 2sin3x cos2x sin x 0


9sin x 6cosx 3sin2x cos2x 8


sin2x 2 cos2x 1 sin x 4cosx



2sin2x cos2x 7 sin x 2cos x 4


sin2x cos2x 3sin x cos x 2



3
2cos x cos2x sin x 0



2
1 cos2x
1 cot2x
sin 2x



44
4 sin x cos x 3 sin 4x 2



33
1
1 sin 2x cos 2x sin 4x
2



tan x 3cotx 4 sin x 3 cos x

. 

33
sin x cos x sin x cos x



44
1
cos x sin x
44



33
4 sin x cos3x 4cos x sin3x 3 3 cos4x 3


2 2 sin x cos x cosx 3 cos2x



2cos x 1 sin x cos x 1



2cos2x 6 cosx sin x


2cos3x 3 sin x cosx 0



sin x 3 cos x sin x 3 cos x 2



2
cos x 3 sin x
cos x 3 sin x 1


sin x cosx cos2x



3
4sin x 1 3sin x 3 cos3x



cos7x cos5x 3 sin2x 1 sin7x sin5x


4sin2x 3cos2x 3 4sin x 1


2
tan x sin2x cos2x 4 cos x
cos x


)

Page - 17 -

2
x
2 3 cos x 2sin
24
1
2cos x 1



2
3cos x 4sin x 6 2 3 3 cos x 4sin x 6




sin x sin2x
3
cos x cos2x



sin 3x 3 cos3x 2sin2x



2
cos x sin2x
3
2cos x sin x 1



1
tan x 3
cos x



3
3sin5x 3 cos15x 1 4 sin 5x

 139

33
5
cos x cos3x sin x sin 3x
8


10cos x 3cotx 4


cos3x sin x 3 cos x sin3x



3
4 sin2x 3cos2x 5cos 3x 0
2



22
x3
4 sin 3 cos2x 1 2cos x , x 0;
24


cos2x 3 sin2x 3 sin x cos x 4 0



3
sin x cos x 2 sin2x 1 sin x cos x 2



22
3 cos x 2 sin x cos x 3 sin x 1 0

nh:

2
1 2sin x cos x 1 sin x cos x



3
4 sin x 1 3sin x 3 cos3x



66
8 sin x cos x 3 3 cos2x 11 3 3 sin 4 x 9sin2x


3 sin2x 2 cos x 1 3cos x 2 cos2x cos3x


Câu 101. 
1 3 9 1 3 5 2
cos x sin x
2 2 2
2 2 2 2

Câu 102. 
1 sin x 1
1 cos x 2

Câu 103. 
3 cos2x sin2x 2sin 2x 2 2

6

Ths.   (Nâng cao)


Page - 18 -
Câu 104. 
2
21
sin 2x 3 sin 2x cos2x 2 sin x
2

Câu 105. 
32
2sin x sin x
4 4 2

Câu 106. 
cos x 3 sin x 2cos x
3

Câu 107. 
sin x 2 sin5x cos x

Câu 108. 
sin x cosx 2 2 sin x cos x

Câu 109. 
sin5x cos5x 2 cos13x

Câu 110. 
cos7x sin5x 3 cos5x sin7x

Câu 111. 
sin 8x cos6x 3 sin6x cos8x

Câu 112. nh:
2
sin x 1 1 cos x cos x

Câu 113. 
5
12cos x 5sin x 8 0
12cos x 5sin x 14

Câu 114. 
xx
2 cos x 3 sin x sin 3 cos
22

Câu 115. 
cos x sin x
cotx tan x
sin x cos x

Câu 116. 
1
sin x tan x cos x
cos x

Câu 117. 
1
4 sin x 3cos x 4 1 tan x
cos x

Câu 118. 
sin5x 3 cos5x 2sin7x

Câu 119. 
3sinx cosx 1

Câu 120. 
sin x 5cosx 1

Câu 121. 
1 3 sin x 1 3 cos x 2 , x 0;

Câu 122. 

13
sin 3x 3 2 cos3x 1 , x ;
99

Câu 123. 
3 2 sin x 3 2 cos x 20

Câu 124. 
sin x 1 sin x cos x 1 cosx

Câu 125. 

22
3cos x sin x sin2x

Câu 126. 
3
3sin3x 3 cos9x 1 sin 3x

Câu 127. 
cos7x cos5x 3 sin2x 1 sin7x sin5x

Câu 128. G
3sin x 4 sin x 5sin 5x 0
3 6 6

Câu 129. 
sin x 3 cos x sin x 3 cosx 2

)

Page - 19 -
Câu 130. 
1
3 sin x cos x 3
3 sin x cos x 1

Câu 131. 
6
3cos x 4 sin x 6
3cos x 4 sin x 1

Câu 132. 
2
1 cos x cos2x cos3x 2
3 3 sin x
3
2cos x cos x 1

Câu 133. 
cos2x 3 sin2x 3 sin x cos x 4 0

Câu 134. 
3
3sin2x 3 cos6x 1 4sin 2x

Câu 135. 
3
cos x 3 sin x 3
3 sin x cos x 1

Câu 136. 
cos9x 2cos6x 2 0

Câu 137. 
sin x sin2x
3
cos x cos2x

Câu 138. 
2
2cos x 3 sin2x 1 3 sin x 3 cos x

Câu 139. 
sin2x cos2x 3sinx cos x 2 0

Câu 140. 
x 5 x 8
2cos sin cos3x 1
22

Câu 141. 
4sin x cos3x 2sin2x 3 sinx cos x

Câu 142. 
2
1 3 cotx
2 3 cotx 1 cot x
sin x sin x

Câu 143. 
3sin x 4 sin x 5sin 5x 0
3 6 6

Câu 144. 
cos2x 3sin2x 5sinx 3cos x 3

Câu 145. 
32
2
4 cos x 2cos x 2 sin x 1 sin2 x 2 sin x cos x
0

2sin x 1

Câu 146. 
3
sin2x cos x 3 2 3 cos x 3 3 cos2x 8 3 cos x sin x 3 3

Câu 147. 
22
sin x 4sin x 3 sin2x 3cos x 2 1 2sin x sin x 3 cos x

Câu 148. 
22
cos x 3 sin2x 1 sin x

Câu 149. 
23
cos x 3 sin2x sin x 1

Câu 150. 
sin3x 3 cos3x sin2x 3 cos2x sinx 3 cosx

Ths.   (Nâng cao)


Page - 20 -


 .

22

a.sin X b.sin X cos X c.cos X d 1 a,b,c,d
.
 
2
cos X
(hay
2
sin X
).
 

2
cos X 0
X k , k Hay X k
sin x 1
2


1

 . Khi

2
cos X 0
X k , k Hay X k
sin x 1
2

1
cho

2
cos X
(hay
2
sin X

22
2 2 2 2
sin X sin X cos X cos X d
1 a. b. c.
cos X cos X cos X cos X


22
a tan X b tan X c d 1 tan X


2
a d tan X b tan X c d 0
.
 
t tanX

 
 

22
1 cos2X 1 cos2X
sin X ; cos X
22


sin2X
sin X cos X
2

1


bsin2X c a cos2X 2d a c

  
sin2X

cos2X

 .


3 2 2 3
4 3 2 2 3 4
a.sin X b.sin X cos X c.sin X cos X d.cos X 0 2
a.sin X b.sin X cos X c.sin X cos X d.sin X cos X e.cos X 3

: 
2
cho
3
cos X
(hay
3

sin X

3
cho

4
cos X
(hay
4
sin X
)



22
cos x 3 sin2x 1 sin x



3 3 2
cos x 4sin x 3cosx sin x sin x 0



4 2 2 4
3cos x 4sin x cos x sin x 0


sin2x 2tan x 3



3
sin x sin2x sin3x 6cos x

)

Page - 21 -


2
cos2x 1
cotx 1 sin x sin2x
1 tan x 2


sin3x cos3x 2cos x 0



3
5sin4x cos x
6sin x 2cos x
2cos2x


3
sin x 4 sin x cosx 0



22
tan x sin x 2sin x 3 cos2x sin x cos x



32
cos x sin x 3sin x cos x 0



2
sin x tan x 1 3sin x cos x sinx 3


22
9
cos 3 2x 3 cos 4x 1 sin 2x , x ;2 1
23



3
2
3
1 cos x
tan x
1 sin x



3 2 2 3
sin x 5sin x cos x 3sin x cos x 3cos x 0



32
cos x sin x 3sin x cos x 0


1 tan x 2 2 sin x



33
sin x cos x sin x cosx



22
3tan x 4 tan x 4 cotx 3cot x 2 0



22
2
3 3sin x x
3 tan x tan x 8 cos 0
42
cos x



33
4cos x 2sin x 3sin x 0



3
6sin x 2cos x 5sin2x cos x


1 3 tan x 2sin2x



3 3 5 5
sin x cos x 2 sin x cos x



44
3sin x 5cos x 3 0



sin x cos x
1
sin2x



3 3 2 2
sin x 3 cos x sin x cos x 3 sin x cosx



23
cot x 2 2 sin x 2 3 2 cos x



3
sin x 2 sin x
4



3
8 cos x cos 3x
3



3
2 sin x 2sin x
4

Ths.   (Nâng cao)

Page - 22 -


3
2 2 cos x 3cos x sin x 0
4



33
4 sin x cos x cos x 3sin x



22
5
4 3 sin x cos x 4 cos x 2sin x
2



2
sin x 3sin x cos x 1 0



4 4 2
1
sin x cos x 2 sin2x cos2x cos 2x

2



4 4 2
3 5 1
sin x cos x sin2x cos2x cos 2x
2 4 4



22
32
sin x 3 sin x cos x 2cos x
2


22
sin x sin2x 3cos x 3


15 17 9
2sin x cos x sin x cos x 3sin 7 x sin x
2 2 2


13 5 3
4 sin 3 x cos x 4 sin x sin x 2 sin x cos x 1
2 2 2

2

22
53
3sin 3 x 2 sin x cos x 5 sin x 0
2 2 2



1
3 sin x cos x
cos x



1
4 sin x 6 cos x
cos x



3
22
7 sin x 2sin2x 3cos x 3 15 0



3 3 2
4sin x 3cos x 3sinx sin x cos x 0


1 3sin2x 2 tan x



x
2 cos x 2 tan
2


cotx tanx 2tan2x

200

7
sin 3x cos3x 2sin x 0 1
2


Câu 151. 
22
sin x 2cos x 3sinx cosx

Câu 152. 
2
sin x 3sinx cos x 1

Câu 153. 
22
2sin x 3cos x cos2x 5sin2x 0

Câu 154. 
3 3 2
4sin x 3cos x 3sin x sin x cos x 0

Câu 155. 
22
97
sin x 3sin x cos x 4 cos 2 x 0
22

)

Page - 23 -
Câu 156. 

22
3
2cos x 3 3 sin x cos x cos 3 x 2 , x 0;
22

Câu 157. 

22
cos 2x 3 sin4x 1 sin 2x , x ;2
3

Câu 158. 

33

cos x 3sin x cos x sin2x 0 , x ;3
4

Câu 159. 
2sin2x 3tanx 5

Câu 160. 
33
sin x 3cos x sinx 0

Câu 161. 
2
cosx sin x 4cosx sin x 0

Câu 162. 
2
sin x tan x 2 3 cos2x sin x cos x

Câu 163. 
2
sin x tan x 1 sin x cos x sin x 1 0

Câu 164. 
11
4 sin x
sin x cos x

Câu 165. 
4 2 4
3cos x sin 2x sin 3 x 0

Câu 166. 
7
sin 3x cos3x 2sin x 0
2

Câu 167. 
3 sin2x 2 cos x 1 2 cos3x cos2x 3cos x

Câu 168. 
3
2cos x sin3x

Câu 169. 
33
1
1 sin 2x cos 2x sin4x
2

Câu 170. 
66
8 sin x cos x 3 3 sin4x 3 3 cos2x 9sin2x 11

Câu 171. 
3 3 2
4sin x 3cos x 3sin x sin x cosx 0



E


 .
a sin x cos x b sin x cos x c 0



2
t1
PP : t sin x cos x, t 2 sin x cos x
2
.
 .
a sin x cos x b sin x cos x c 0



2
1t
PP : t sin x cos x, t 2 sin x cos x
2
.
 .
22
a tan x cot x b tan x cotx c 0


sin x 0
k
ÐK : sin2x 0 x , k
cos x 0

2
.


2 2 2
PP : t tan x cotx , t 2 tan x cot x t 2
.
Ths.   (Nâng cao)


Page - 24 -
 .
22
a tan x cot x b tan x cotx c 0


sin x 0
k
ÐK : sin2x 0 x , k
cos x 0
2
.


2 2 2
PP : t tan x cotx , t 2 tan x cot x t 2
.
 .
44
a sin x cos x bsin2x c 0

4 4 2 2
11
PP : t sin2x, t 1 sin x cos x 1 sin 2x 1 t
22
.
 .
44
a sin x cos x bcos2x c 0



4 4 2 2 2
1 1 1 1 1
PP : t cos2x, t 1 sin x cos x 1 sin 2x cos 2x t
2 2 2 2 2
.
 .
66
a sin x cos x bsin2x c 0



6 6 2 2
33
PP : t sin2x, t 1 sin x cos x 1 sin 2x 1 t
44
.

 .
66
a sin x cos x bcos2x c 0



6 6 2 2 2
3 1 3 1 3
PP : t cos2x, t 1 sin x cos x 1 sin 2x cos 2x t
4 4 4 4 4
.
 .
44
a sin x bcos x ccos2x d 0



2
24
2
2
4
1t
1 cos2x 1 t
sin x sin x
2 2 4
PP : t cos2x, t 1
1 cos2x 1 t
1t
cos x

cos x
22
4


201

23
sin x sin x cos x 0

202

33
3
sin x cos x 1 sin2x
2

203
2 sin x cos x tan x cotx

204
3 cotx cosx 5 tan x sin x 2

205

32
2
3 1 sin x
x
3 tan x tan x 8 cos

42
cos x

206:

33
2sin x sin x 2cos x cos x cos2x

207

2 3 4 2 3 4
sin x sin x sin x sin x cosx cos x cos x cos x

208

2 3 3
tan x 1 sin x cos x 1 0

209

3
2
cos 2x 2 sin x cosx 3sin2x 3

)

Page - 25 -
210
2sin x cotx 2sin2x 1

211
cos2x 5 2 2 cosx sin x cos x

212

33
cos x sin x cos2x

213

33
cos x sin x 1

214

22
2cos2x sin x cosx sin x cos x 2 sin x cos x

215

22
3tan x 4 tan x 4 cotx 3cot x 2 0

216

2 3 2 3
tan x tan x tan x cotx cot x cot x 6

217

2
2
2
2 tan x 5 tan x 5 cotx 4 0
sin x

218

2
2
15
cot x tan x cotx 2 0
2
cos x


219

33
1 cos x sin x sin x

220

32
cos x cos x 2sin x 2 0

221
cos2x 5 2 2 cos x sin x cosx

222

cotx tan x sin x cos x

223
1 tanx sin x cosx

224
sin2x 2 sin x 1
4

225
sin2x 12 sin x cosx 12 0

226

sin x cos x
1
sin2x 1

227nh:

3
3
1 cos2x 1 cos x
1 cos2x
1 sin x

228
5 sin x cos x sin3x cos3x 2 2 2 sin2x

229

1 sin x cos x sin2x 2cos2x 0

230

22
sin x cos x cos2x sin x cos x sin x cosx

231
cos2x 5 2 2 cos x sin x cosx

232

23
cos x sin x cos x 0

233

3
4 sin x 1 3sin x 3 cos3x

234

22
1 sin x cos x 1 cos x sin x 1 sin2x

235
sin x cos x 2sin x 2cosx 2

236
1 tan x 2 2 sin x

237
2 cot2x cot3x tan2x cot3x