Bài tập phương trình lượng giác

Đỗ Ngọc Nam_THPT Trung Giã 
BÀI TẬP PHƯƠNG TRÌNH LƯỢNG GIÁC 
[1] sin sin 5 8cos .cos3
sin 3 sin
x x
x x
x x
+ = . Đ/s: ( ) ( );
8 4 2
l
x l x k kpi pi pi pi= + ∈ = + ∈Z Z 
[2] 2sin 6 2sin 4 3 os2 3 sin 2x x c x x− + = + . Đ/s: x kpi= , ;
12 2 18 3
k k
x x
pi pi pi pi
= − + = + 
[3] 4sin3x -13sin2x + 4sinx = 3cos3x – 13cosx + 8cos2x. Đ/s: 2 , 2 , 2
6 3 2 2
k
x x k x kpi pi pi pipi α pi= + = + = − + 
[4] ( )2 cos sin1
tan cot 2 cot 1
x x
x x x
−
=
+ −
. Đ/s: ( )2
4
x k kpi pi= − + ∈ℤ 
[5] 1 sin 2cot 2sin
sin cos 22
x
x x
x x
pi 
+ = + +  
. Đ/s: 52 ; 2 .
6 6
x k x kpi pipi pi= + = + 
[6] 2)cos3(sin3sin =+ xxx . Đ/s: 
6
kpi pi+ 
[7] ( ) ( )3sin 2 cos 3 2 3 cos 3 3 cos 2 8 3 cos s inx 3 3 0x x x x x+ − − + − − = . Đ/s: ,3
2
x k
k
x k
pi
pi
pi

= + ∈ Ζ

=
[8] 24sin .sin .sin 4 3.cos .cos .cos 2
3 3 3 3
x x x x x x
pi pi pi pi       
+ − − + + =       
       
. Đ/s: 2 , Z
18 3
x k kpi pi= + ∈ 
[9] ( ) sin3sin2 cos2 tan sin cos
cos
x
x x x x x
x
− + = + 
[10] 4 2 43sin 2cos 3 3 3 cos 1x x cos x cos x x+ + = − + . Đ/s: & 2 ,
4 2
x k x k kpi pi pi= + = ∈Z 
[11] 52 2 os sin 1
12
c x x
pi 
− = 
 
. Đ/s: 3;
6 4
x k x kpi pipi pi= + = + 
[12] 2 4 23sin 4 .sin os 1 os
2
x x c x c x+ − = . Đ/s: vô nghiệm 
[13] sin4x + cos3x + cosx = 4sinx + 2 
[14] 2sin(x +
4
pi ).cos(x + 
6
pi ) + cos2x = 0. Đ/s: 7,
4 24
x k x kπ ππ π=− + = + 
[15] 2 sin(2 ) sinx 3cos 2 0
4
x x
pi
+ − − + = 
[16] 2tanx + cotx = 2sin2x + 
x2sin
1
. 
[17] cos8x + 3cos4x + 3cos2x = 8cosx.cos33x – 
2
1
. 
[18] sin 2x.(1 + tanx) = 3sinx.(cosx – sinx) + 3 
[19] cot2x – 2tan4x – tan2x = - 4 3 
[20] 2sin2(x - 
4
pi ) = 2sin2x - tanx. 
[21] 
xx
xx
cossin
cossin
−
+
 + 2tan2x + cos2x = 0. 
[22] sin 2 1 2 os
sin cos 2. tan
x
c x
x x x
+ =
+
. Đ/s: pipi kx +=
2
; .,,
3
2
4
Z∈+= tktx pipi 
Đỗ Ngọc Nam_THPT Trung Giã 
[23] ( ) 32cos13 3 cos3 cos5 8cos .cos 4x x x x x+ + = . Đ/s: 
12
kπ
[24] sin 2 cos 2 tan cot
cos sin
x x
x x
x x
+ = − . Đ/s: 2
3
kπ π± + 
[25] sin 4 2 cos3 4sin cosx x x x+ = + + . Đ/s: 52 ; 2 ; 2
6 6
k k kπ ππ π π+ + 
[26] 
3 3sin cos 12 2 cos
2 sin 3
x x
x
x
−
=
+
. Đ/s: 2
2
kπ π+ 
[27] ( ) 22cos3 .cos 3 1 sin 2 2 3 cos 2
4
x x x x
π + + = +   
[28] 5cos sin 3 2 sin 2
4
x x x
π + − = +   
. Đ/s: 2
3
kπ π± + 
[29] 4 4 5cos sin sin 2 .cos 2 tan tan
4 4 4 4
x x x x x x
π π π         + + = − + −             
. Đ/s: ;
8 2
kk π ππ − + 
[30] 1 sin 2cot 2sin
sin cos 22
x
x x
x x
π + = +   +
. Đ/s: 2,
2 4 3
lkπ π ππ+ + 
[31] ( ) 2sin 2 1 2cos3 sin 2sin 2 0
4
x x x x
π + + − + =  
. Đ/s: 2
2
lπ π+ 
[32] 22 3 cos 2 tan 4sin cot 2
4
x x x x
π − = − +  
. Đ/s: 7,
12 36 3
kkπ π ππ+ + 
[33] 2 22 sin sin 3 sin . tan tan
3 3 4 4
x x x x x
π π π π               + + − = + + −                     
. Đ/s: 5, 2 , 2
6 6
k k kπ ππ π π+ + 
[34] 2 2 2
1 1
sin 2 4sin 2
cos sin 4
x x
x x
π     − =− −       
. Đ/s: 
4 2
kπ π+ 
[35] ( ) 21 2cos3 sin sin 2 2sin 2 0
4
x x x x
π + + − + =  
. Đ/s: 2
2
kπ π+ 
[36] 2sin ( 3 sin ) 2 3 3 0
2sin 1
x x cosx cos x
x
+ − −
=
−
. Đ/s: 2 , 5
6 5
k k tπ π+ ≠ 
[37] 15tan cot 4sin
4
x x x
π − = +   
. Đ/s: 2,
4 4 3
k kπ π ππ+ + 
[38] 32cos cos 2 sin 0x x x+ + = . Đ/s: , 2
4 2
k kπ ππ π− + + 
[39] 2 2 24sin 1 8sin .cos 4cos 2x x x x+ = + 
[40] 2 22sin .sin 2 cos .sin 2 1 2cos
4
x x x x x
π − + = −   
[41] sin 2 2cos 2 1 sin 4cosx x x x+ = + − . Đ/s: 2
3
x k kpi pi= ± + ∈ℤ 
Đỗ Ngọc Nam_THPT Trung Giã 
[42] (1 cos 2 )sin 2 2(sin 3 sin )(1 sin )
1 sin
x x
x x x
x
+
= + +
−
. Đ/s: 2 
2
2 2
3
x k
x k k Z
x k
pi
pi
pi
pi
pi

 =

− = + ∈


 = ± +

[43] 1 3cos cos 2 2 cos 3 4 sin .sin 2x x x x x+ + − = . Đ/s: 2
2 2
3
x K
x K
pi
pi
pi
pi

= +


= ± +

[44] 4sin3x -13sin2x + 4sinx = 3cos3x – 13cosx + 8cos2x. Đ/s: (sin2x - cosx)(4cosx +3sinx – 5) = 0???? 
[45] 





+=+
4
3sinsin222sinsin2 2 pixxxx . Đ/s: x = k
28
,
pipi
pi kx +−= . 
[46] 2tancot)
4
2(cos2 2 −−=+ xxx pi . Đ/s: ,
8 2
l
x l
π π
= + ∈ ℤ 
[47] 1 cos 2sin 2 3 cos 3
sin
x
x x
x
+
= + − . Đ/s: 2
3
kπ π± + 
[48] 2cos 2 1cot 1 sin sin 2
1 tan 2
x
x x x
x
− = + −
+
. Đ/s: 
4
lπ π+ 
[49] 
4 4 2
2 2 2sin cos sin 2 cot 2 .cos 2 cot 2 cos
1 cos 2
x x x
x x x x
x
+ +
− = +
−
. Vô nghiệm 
[50] 4sin 2sin 2 1
3 6
x x
π π     + − − =       
. Đ/s: ;
3
l lππ π− + 
[51] ( )2sin 3 cos .cos 2 tan tan 2x x x x x= + . Đ/s: lπ 
[52] 32 2 cos 2 sin 2 .cos 4sin 0
4 4
x x x x
π π     + + − + =       
. Đ/s: 2 ; 2
2
l lππ π− + 
[53] cos 2 3sin 2 5sin 3cos 3x x x x+ + − = 
[54] 2cos 4 2cos sin 3 sin 1
3 3
x x x x
π π     + + − + − =       
. Đ/s: 2; 2 ;
2 6 30 5
lk lπ π π ππ π+ + − + 
[55] 217sin 2 16 2 3 sin .cos 20sin
2 2 12
x
x x x
π π     + + = + +       
. Đ/s: 52 ; 2
2 6
l lπ ππ π+ − + 
[56] 1sin .cos 2 sin 2 .cos3 sin 5
2
x x x x x= − 
[57] )
2
sin(2
cossin
2sin
cot
2
1 pi
+=
+
+ x
xx
x
x . Đ/s: pipi kx +=
2
; .,,
3
2
4
Z∈+= tktx pipi 
[58] ( )4 44cos 2 sin cos 3 sin(2 ) cos(2 )3 3x x x x x
pi pi
+ = + + + . Đ/s: ( )
4 2
x k k Zpi pi= + ∈ 
[59] ( )3 sin 2 sin cos 2 cos 2x x x x+ + − = 
[60] 2cosx + tanx = 1 + 2sin2x. 
[61] 83cot tan 8sin
3
x x x
pi 
− = − 
 
[62] 2 cos 12(1 sin )(tan 1)
sin cos
x
x x
x x
−
+ + =
+
Đỗ Ngọc Nam_THPT Trung Giã 
[63] 2 22sin 2sin t anx
4
x x
pi 
− = − 
 
[64] 2 os6x+2cos4x- 3 os2x = sin2x+ 3c c 
[65] 22 os3x.cosx+ 3(1 s in2x)=2 3 os (2 )
4
c c x
pi
+ + 
[66] 52 2 os sin 1
12
c x x
pi 
− = 
 
[67] 2
3 4 2sin 2 2 3 2(cot 1)
sin 2cos
x
x
xx
+
+ − = + . Đ/s: 
6 2
x kpi pi= + 
[68] ( ) ( )
2cos . cos 1
2 1 sin .
sin cos
x x
x
x x
−
= +
+
 Đ/s: 2
2
x kpi pi= − + và 2x mpi pi= + ( ),k m∈Z 
[69] cos2x 2sin x 1 2sin x cos 2x 0+ − − = . Đ/s: x kpi= ; 2
6
x kpi pi= + hoặc 5 2
6
x kpi pi= + , k Z∈ 
[70] 0
10
5cos3
6
3cos5 =





−+





+
pipi
xx . Đ/s: )()
3
2
arccos(
2
1 Zkkx
kx
∈





+−±=
=
↔
pi
pi
[71] 





−=−+
24
cos2sin
2
cossin
2
sin1
22 x
x
x
x
x pi
. Đ/s: lπ 
[72] 2 cos5 .cos 3 sin cos8 x x x x+ = . Đ/s: 72 ; 2 ; 2 , ( )
2 6 6
x k x k x k k Zpi pi pipi pi pi= + = − + = + ∈ 
[73] 23tan 3 cot 2 2 tan
sin 4
x x x
x
+ = + 
[74] ( )6 68 sin 3 3 sin 4 3 3 2 9sin 2 11x cos x x cos x x+ + = − + 
[75] 2cos 3 tan 2 0x x+ + = . Đ/s: 2 ; 2
3
k kππ π π+ − + 
[76] ( )( )2tan 3 1 2 sin tan 2 cosx x x x+ = + + (đề 21) 
[77] ( ) ( )2cos 2 1 cos sin 2 sin 3 sin cosx x x x x x− − = + (đề 20) 
[78] 
2
cos 4 sin 2 2 2 sin 3
cos3 sin 3 4
x x
x
x x
pi+   
= + +   +   
 (đề 16) 
[79] sin sin sin 4 sin 2
3 3
x x x x
pi pi   
+ + + = −   
   
 (đề 12) 
[80] ( )2 3cos 2 cos 4 tan 2 .cot 1
4
x x x x+ − = − (đề 9) 
[81] cos cos 7 3 3 sinx x x− = (đề 8) 
[82] sin 2 cos 2 2 2 sin 0
4 4
x x x
pi pi   
− − − =   
   
 (đề 7) 
[83] ( )( )22 sin 1 sin 2 3sin 1 sin 4 .cosx x x x x+ − + = (đề 6) 
[84] ( ) ( )
22 sin cos 1 2sin 2
1 tan
sin 3 sin 5
x x x
x
x x
− +
= −
+
 (đề 5) 
[85] ( )2sin 3 cos .cos 2 tan 2 tanx x x x x= + (đề 4) 
Đỗ Ngọc Nam_THPT Trung Giã 
[86] 2cos tan 1 2sin 2x x x+ = + (đề 2) 
[87] ( )3 sin 2 sin cos 2 cos 2x x x x+ + − = (đề 1) 
[88] 2sin .sin 4 2 2 cos 4 3 cos .sin .cos 2
6
x x x x x x
pi 
= − − 
 
 (đề 19) 
[89] sin 3 2cos3 cos 2 2sin 2 2sin 1 0x x x x x+ + − − − = (đề 18) 
[90] 22cos 2 3tan 1 3 tan .sin
3
x x x x
pi 
+ + = + 
 
 (đề 1.2011) 
[91] 32 2 sin cos cos 2sin 2 3
8 2 8 2
x x
x x
pi pi    
− − − = −    
    
 (đề 2.11) 
[92] 
6 4
2
16cos 2cos
sin
54 51cos
x x
x
x
+
=
−
 (đề 3.11) hàm số đưa về cos2x. Đ/s: 52 ; 2
6 6
k kpi pipi pi+ + 
[93] 
2
22 sin 1 tan
cos 2 4cos 3 2 2
x x
x x
−
=
+ +
. Đ/s: 
2
kpi pi+ 
[94] ( )24sin tan 2 1 tan sin 3 1x x x x+ + + = . Đ/s: 2 3; ; 2
4 20 5 4
kk kpi pi pi pipi pi− + + + 
[95] 1 12 sin 2 4sin 1
sin 6 2sin
x x
x x
pi   
− − = − −   
   
[96] 
2
2
2cos 2cos 3 4 3 sin 0
sin
2
x x
x
x
+ −
+ = . Đ/s: 2 ; 2
3 6
k kpi pipi pi+ + 
[97] tan tan 2 tan 3 tan 4 0x x x x+ + + = 
[98] 8sin tan cot 4cot 2
6
x x x x
pi 
+ + + = 
 
[99] ( ) 23 sin 2 3sin 2cos 3cos 5x x x x− = + − 
[100] ( )3 sin 2 1 2cos cos3 1
1 2cos cos 2
x x x
x x
+ +
=
+ +
[101] 
3
4 2
4
x
x
x x
pi
pi
 
+ 
  
= + +  
sin
cot
sin cos
[102] 4 4 2 2
1
sin cos cos 0
4sin 2
x x x
x
+ − + = . Vô nghiệm. 
[103] 1sin cos 2 2cos .sin
2 4 2
x
x x x
pi 
− + − = 
 
. Đặt 
2 4
x
t
pi
= − 
[104] ( ) ( )2 2sin9 cos sin8 cos8 sin2 cos2 0
2 2
x x x x x x+ − − + − = 
[105] ( )2 2sin9 cos sin8 cos8
2 2
x x x x+ − − = 
[106] 
3 3
sin cos sin cos
sin3 cos3 sin cos
x x x x
x x x x
− −
=
− +
Đỗ Ngọc Nam_THPT Trung Giã 
Bài tập trong đề 
Bài 1 2 2 2sin tan cos 0
2 4 2
x x
x
pi 
− − = 
 
Bài 2 ( )( )2cos 1 2sin cos sin 2 sinx x x x x− + = − 
Bài 3 4 4 3cos sin cos sin 3 0
4 4 2
x x x x
pi pi   
+ + − − − =   
   
Bài 4 cos3 cos 2 cos 1 0x x x+ − − = 
Bài 5 ( )2sin cos 3 cos 2 2x x x+ − = 
Bài 6 ( )2sin 1 cos 2 sin 2 1 2cosx x x x+ + = + 
Bài 7 3 cos5 2sin 3 .cos 2 sin 0x x x x− − = 
Bài 8 sin 2 cos 2 3sin cos 1 0x x x x− + − − = 
Bài 9 2 2 2 2sin 3 cos 4 sin 5 cos 6x x x x− = − 
Bài 10 2cot tan 4sin 2
sin 2
x x x
x
− + = 
Bài 11 ( ) 25sin 2 3 1 sin tanx x x− = − 
Bài 12 1 sin cos sin 2 cos 2 0x x x x+ − − − = 
Bài 13 cot sin 1 tan . tan 4
2
x
x x x
 
+ + = 
 
Bài 14 22cos 2 sin 9 sin 1x x x− = + 
Bài 15 3 3 2 2sin 3 cos sin .cos 3 sin .cosx x x x x x− = − 
Bài 16 ( )3sin cos .sin 2 3 cos3 2 cos5 sinx x x x x x+ − = + 
Bài 17 ( )sin 2 cos 2 cos 2 cos 2 sin 0x x x x x+ − − = 
Bài 18 sin 3 cos35 sin cos 2 3
1 2sin 2
x x
x x
x
+ 
+ = + + 
Bài 19 2cos 2 1cot 1 sin sin 2
1 tan 2
x
x x x
x
− = + −
+
Bài 20 2 2cos 3 .cos 2 cos 0x x x− = 
Bài 21 ( )6 62 sin cos sin .cos 0
sin cos
x x x x
x x
+ −
=
−
Đỗ Ngọc Nam_THPT Trung Giã 
Bài 22 ( ) ( )2 21 sin cos 1 cos sin 1 sin 2x x x x x+ − + = − 
Bài 23 1 1 34sin
sin 4
sin
2
x
x
x
pi
pi
 
− = + 
   + 
 
Bài 24 ( )( )( )
1 2sin .cos
3
1 2sin 1 sin
x x
x x
−
=
+ −
Bài 25 
( )1 sin cos 2 sin
cos4
1 tan 2
x x x
x
x
pi 
+ + − 
 
=
−