phương trình lượng giác ôn thi ĐH

Bài tập ôn tập Phương trình lượng giác()_2010-2011
CÁC DẠNG PHƯƠNG TRÌNH LƯỢNG GIÁC
1. Biến đổi thành phương trình chứa một hàm số
lượng giác (Bậc nhất, hai, ba....)
1.DBA06.
3 3
2 3 2
cos3 cos sin 3 sin
8
x x x x
+
− =
16 2
k
π π
± +
2.DBD07.
2 2 sin cos 1
12
x x
π
− =
 
 ÷
 
;
4 3
x k x k
π π
π π
= + = +

3B06
cot sin 1 tan tan 4
2
x
x x x
 
+ + =
 ÷
 
π 5π
+kπ; +kπ
12 12
4.A_2005
2 2
cos 3 cos 2 cos 0x x x− =
( )
2
π
= ∈
x k k Z
5.D05.
4 4
3
cos sin cos sin 3 0
4 4 2
x x x x
   
+ + − − − =
 ÷  ÷
   

π π
4
x k
π
π
= +
6.B04.
2
5sin 2 3(1 sin ) tanx x x− = −
π 5π
+ k2π; +k2π
6 6
7.B_2003.
2
cot tan 4sin 2
sin 2
x x x
x
− + =

π
x=± +kπ
3
8.A_2002 Tìm nghiệm
(0;2 )x∈ π
của pt:
cos3 sin 3
5 sin cos 2 3
1 2sin 2
x x

x x
x
+
 
+ = +
 ÷
+
 
.
5
;
3 3
x x
π π
= =
9.DB _2002
4 4
sin cos 1 1
cot 2
5sin 2 2 8sin 2 .
x x
x
x x
+
= −
6
k
π
π
± +

10.DBA 03
( )
2
cos 2 cos 2 tan 1 2x x x+ − =
2 , 2
3
k k
π
π π π
± + +
11.A_06
6 6
2(cos sin ) sin cos
0
2 2sin
x x x x
x
+ −
=
−
5π
x = + 2kπ
4
12.D_2006
cos3 cos 2 cos 1 0x x x
+ − − =
2
; 2
3
x k x k

π
π π
= = ± +
13.D02 Tìm
[ ]
0;14x∈
cos3 4cos 2 3cos 4 0x x x− + − =
.
3 5 7
; ; ;
2 2 2 2
x x x x
π π π π
= = = =
14.DB_2008
2
3sin cos 2 sin 2 4 sin cos
2
x
x x x x+ + =

7
2 , 2 , 2
2 6 6
k k k
π π π
π π π
−
+ + +
15.DB.D_2008

4 4
4(sin cos ) cos 4 sin 2 0x x x x+ + + =
2
2
x k
π
π
= − +
16.DBB.03
6 2
3cos 4 8cos 2cos 3 0x x x− + + =
,
4 2
k k
π π
π
+
17.
3 3
sin .sin 3 cos cos3 1
8
tan tan
6 3
π π
+
= −
   
− +
 ÷  ÷
   

x x x x
x x
6
π
π
= − +
x k

18.
3 3
sin .(1 cot ) cos (1 tan ) 2sin 2+ + + =x x x x x

2
4
π
π
= +x k
19.2tanx + cotx =
2
3
2sin x
+

;( )
3
x k k
π
π
= + ∈
¢

20.
2 3
10 2 4 6 3 . 8 . 3cos x cos x cos x cosx cosx cosx cos x+ + = +
2x k
π
=
21.
6 6
4sin x cos x cos x+ =
;
2
k
x
π
=
22.
5 7
sin 2 3cos 1 2sin
2 2
x x x
π π
   
+ − − = +
 ÷  ÷
   
,
x∈
(π/2;3π)
1 2 3 4 5
13 5 17

; 2 ; ; ;
6 6 6
x x x x x
π π π
π π
= = = = =
23.
2 2
2 1
os os (sinx 1)
3 3 2
C x C x
π π
   
+ + + = +
 ÷  ÷
   

5
2 , 2 , 2
6 6
x k x k x k
π π
π π π
= = + = +
24.
( )
4 4
2
1 cot 2 cot

2 sin cos 3
cos
x x
x x
x
+
+ + =
4 4
k
x
π π
= +
25.
4 4
4
sin 2 os 2
os 4
tan( ).tan( )
4 4
x c x
c x
x x
π π
+
=
− +
2
k
π
26.A-10

(1 sin x cos 2x)sin x
1
4
cos x
1 tan x
2
π
 
+ + +
 ÷
 
=
+

7
2 , 2
6 6
x k x k
π π
=− + π = + π
27.
6 3 4
8 2 cos 2 2 sin sin 3 6 2 cos 1 0x x x x+ − − =

8
π
x kπ= ± +
28.
( )
2 cos sin

1
tan cot 2 cot 1
x x
x x x
−
=
+ −
2
4
x k
π
π
= − +
29.
( )
4 4
sin cos 1
tan cot
sin 2 2
x x
x x
x
+
= +
VN
2. Phương trình bậc nhất với sin và cos
Bài tập ôn tập Phương trình lượng giác()_2010-2011
1.D_07
2
sin cos 3 cos 2

2 2
x x
x+ + =
 
 ÷
 
π π
+k2π; - +k2π
2 6
2.CĐ_2008
sin 3 3 cos3 2sin 2x x x− =
( )
4 2
2 , ,
3 15 5
x k x k k
π π π
π
= + = + ∈
Z
3.D_2009
3 cos5 2sin 3 cos 2 sin 0x x x x− − =
x k
18 3
π π
= −
hay
x k
6 2
π π

= − −
4.B0
3
sin cos sin 2 3 cos3 2(cos 4 sin )x x x x x x+ + = +
2
x k2 , x k
6 42 7
π π π
=− + π = +
5.A_2009
(1 2sin ) cos
3
(1 2sin )(1 sin )
x x
x x
−
=
+ −
2
18 3
= − +
x k
π π
6.DB _03
( )
2
2 3 cos 2sin
2 4
1
2 cos 1

x
x
x
 
− − −
 ÷
 
=
−
π
3
k
π
π
+
7.DB_A_
cos sin cos (sin cos )x x x x x
+ + = +
2
2 2 3 1 3 3
2
3
π
π
= +
x k
8.DB_A_06.
2sin 2 4sin 1 0
6
x x

π
 
− + + =
 ÷
 
7π
x= +k2π; x=kπ
6
9.
3cos sin 2 3 cos 2 3 sinx x x x− = +
9.DB-D _2004
( )
sin sin 2 3 cos cos 2x x x x+ = +
2 / 9 2 / 3;.. 2
π π π π
= + = +
x k x k
10.DBA_2005Tìm n
o
trên
(0; )π
của
2 2
3
4sin 3 cos 2 1 2 cos
2 4
x
x x
 
− = + −

 ÷
 
π
.
5π 17π 5π
; ;
18 18 6
11.
2sin 5 3 os3 sin 3 0x c x x+ + =
2
,
24 4 3
k
x x k
π π π
π
=− + = −
12.
2 2
2cos 2x 3cos4x 4cos x 1
4
π
− + = −
 
 ÷
 
k
k ,
12 36 3
π π π

+ π +
3.Biến đổi thành phương trình tích
1.B-10. (sin 2x + cos 2x) cosx + 2cos2x – sin x = 0
x =
4 2
k
π π
+

2.D-10
sin 2 cos 2 3sin cos 1 0x x x x
− + − − =
5
2 , 2
6 6
x k x k
π π
= + π = + π
3.A_2008
1 1 7
4sin
3
sin 4
sin
2
x
x
x
π
 

+ = −
 ÷
π
 
 
−
 ÷
 
5
; ;
4 8 8
x k x k x k
π π π
π π π
=− + =− + = +
4.B_08.
3 3 2 2
sin 3 cos sin cos 3 sin cosx x x x x x− = −
;
4 2 3
k
x x k
π π π
π
= + =− +
5.D_2008
2sin (1 cos 2 ) sin 2 1 2cosx x x x+ + = +

2
; 2

4 3
x k x k
π π
π π
= + = ± +
6.A_07.
2 2
(1 sin ) cos (1 cos )sin 1 sin 2x x x x x+ + + = +
π π
x = - + kπ; x = + k2π; x = k2π
4 2
7.B_2007.
2
2sin 2 sin 7 1 sinx x x+ − =
2 5 2
2 ; ;
8 18 3 18 3
x k x k x k
π π π π π
π
= + = + = +
8.B_2005.
1 sin cos sin 2 cos 2 0x x x x+ + + + =
2
; 2
4 3
x k x k
π π
π π
=− + =± +

9.D_2004
(2cos 1)(2sin cos ) sin 2 sinx x x x x− + = −
π π
x = ± + k2π; x = - + kπ
3 4
10.A03.
2
cos 2 1
cot 1 sin sin 2
1 tan 2
x
x x x
x
− = + −
+
4
k
π
π
+
11.D03
2 2 2
sin tan cos 0
2 4 2
x x
x
π
 
− − =
 ÷

 
π
π + k2π; - +kπ
4
12.B_02
2 2 2 2
sin 3 cos 4 sin 5 cos 6x x x x− = −
π π
,
2 9
k k
13.DB.A08
2
tan cot 4 cos 2x x x= +
,
4 2 8 2
k k
π π π π
+ − +
14.DB.A08.
2
sin 2 sin
4 4 2
x x
π π
− = − +
   
 ÷  ÷
   
, 2

4 3
k k
π π
π π
+ ± +
Bài tập ôn tập Phương trình lượng giác()_2010-2011
15.DB.B_2008
1
2sin sin 2
3 6 2
x x
π π
   
+ − − =
 ÷  ÷
   
x k ,x k
3 2
π π
π π
=− + = +
16.DB.A07
1 1
sin 2 sin 2cot 2
2sin sin 2
x x x
x x
+ − − =
x k
4 2

π π
= +
17.DB.B07
5 3
sin cos 2 cos
2 4 2 4 2
x x xπ π
   
− − − =
 ÷  ÷
   
2
; 2 ; 2
3 3 2
x k x k x k
π π π
π π π
= + = + = +
18.DB.B07
sin 2 cos 2
tan cot
cos sin
x x
x x
x x
+ = −
2
3
x k
π

π
= ± +
19.DB.D07
(1 tan )(1 sin 2 ) 1 tanx x x− + = +
π
kπ;- +kπ
4
20.DB.B06
2 2 2
(2 sin 1) tan 2 3(2 cos 1) 0x x x− + − =
π π
± +k
6 2
21B_2006
( ) ( )
cos 2 1 2cos sin cos 0x x x x+ + − =
π π
x + kπ; + k2π; π + k2π
4 2
22.DB.06
3 3 2
cos sin 2sin 1x x x+ + =
; 2 ; 2
4 2
k k k
π π
π π π
− + − +
23.DB.D_2006
3 2

4sin 4sin 3sin 2 6cos 0x x x x+ + + =
π 2π
x = - + k2π; x = ± + k2π
2 3
24.DBD05.
3 sin
tan 2
2 1 cos
x
x
x
π
 
− + =
 ÷
+
 
π 5π
+ k2π; + k2π
6 6
25.DB.B _2004
1 1
2 2 cos
4 sin cos
x
x x
 
+ + =
 ÷
 

π
26.DB.D03
( )
( )
2
cos cos 1
2 1 sin
sin cos
x x
x
x x
−
= +
+

2 , 2
2
k k
π
π π π
− + +
27.DB.D _2003
2cos 4
cot tan
sin 2
x
x x
x
= +

2
3
k
π
π
± +
28.DBA 02.
( )
2
2
tan cos cos sin 1 tan tan
x
x x x x x
+ − = +
2k
π
29.
( )
2
4
4
2 sin 2 sin 3
tan 1
cos
x x
x
x
−
+ =

2 5 2
,
18 3 18 3
k k
π π π π
+ +

30.DBA03
( )
3 tan tan 2 sin 6 cos 0x x x x− + + =
3
k
π
π
± +
31.
3cos sin 2 3 cos 2 3 sinx x x x− = +

, 2 , 2
3 6 2
x k k k
π π π
π π π
= + + +
32.DB.D_2005
sin 2 cos 2 3sin cos 2 0x x x x+ + − − =

π π 5π
x = + k2π; x = π + k2π; x = ; x = +k2π
2 6 6

33.9sinx + 6cosx – 3sin2x + cos2x = 8
2
2
k
π
π
+
34.
x x x x
3
2 2 cos2 sin 2 cos 4sin 0
4 4
π π
   
+ + − + =
 ÷  ÷
   

x k
4
π
π
= − +
;
x k x k
3
2 ; 2
2
π
π π

= = +
35.
sin .tan 2 3(sin 3 tan 2 ) 3 3+ − =x x x x

6 2
k
π π
− +

36.
sin 3 sin 2 sin
4 4
π π
   
− = +
 ÷  ÷
   
x x x

4
x k
π
π
= ± +
37.
tan tan .sin 3 sin sin 2
6 3
π π
   
− + = +

 ÷  ÷
   
x x x x x
2
, 2
2 3
k
k
π π
π
− +
38.
3 3 2
4 3 . 0cos x sin x cosx sin x sinx− − + =
π π
π π
− + ± +
,
4 6
k m
39.
( )
2
2 sin cos
4
π
 
− = −
 ÷
 

x x tg x

2
, 2
4 3
k k
π π
π π
+ ± +

40.sin
2
3x - sin
2
2x - sin
2
x = 0
,
6 3 2
k
k
π π π
+
41
( )
( )
3
sin 2 cos 3 2 3 os 3 3 os2 8 3 cos s inx 3 3 0x x c x c x x
+ − − + − − =
, 2

3
x k x k
π
π π
= + =
42.
2 sin 2 3sin cos 2
4
x x x
π
+ = + +
 
 ÷
 
2 , 2
2
k k
π
π π π
− + +
43.
)
2
sin(2
cossin
2sin
cot
2
1
π

+=
+
+
x
xx
x
x
π
π
kx
+=
2
;
3
2
4
ππ
t
x
+=
44.
(
)
2 2
2 sin sin 2 cos sin 2 1 2 cos
4
x x x x x
π
− + = −
2

2
x k
π
= + π
Bài tập ôn tập Phương trình lượng giác()_2010-2011
45.
2
(1 sinx) cosx+ =
x k2 ,x k2
2
p
= p = - + p
46.
2 2
2 sin 2sin tan
4
x x x
π
 
− = −
 ÷
 
4
k
π
π
+
47.cos
3
x+cos

2
x+2sinx–2 = 0
2 ; 2
2
x k x n
π
π π
= = +
Phương trình đẳng cấp
1.DBA_04.
3 3
4(sin cos ) cos 3sinx x x x+ = +
,
4 3
k k
π π
π π
+ ± +
2.
3 3 2 2
sin 3 cos sin cos 3 sin cosx x x x x x− = −
;
4 2 3
k
x x k
π π π
π
= + =− +
3.DBA_2005
3

2 2 cos 3cos sin 0
4
x x x
 
− − − =
 ÷
 
π
π π
x= +kπ; x= +kπ
2 4
4.cosx = 8sin
3
6
x
π
 
+
 ÷
 
x = k
π
5.tanx.sin
2
x−2sin
2
x=3(cos2x+sinx.cosx)
; 2
4 3
x k x n

π π
π π
=− + =± +
6.sinx−4sin
3
x+cosx =0
4
x k
π
π
= +
.
7.
3 3 2
4sin 3cos 3sin sin cos 0x x x x x+ − − =
,
4 3
k k
π π
π π
+ ± +
8.
2 2 tan 3Sin x x
+ =
4
x k
π
π
= +
9.

2 2
os 3 sin 2 1 sinC x x x− = +
,
3
k k
π
π π
− +
,
10.
4 2 2 4
3cos 4sin cos sin 0x x x x− + =
,
4 3
x k x k
π π
π π
=± + =± +
Giải bằng phương pháp đặt ẩn phụ
hoặc góc phụ
1.
3
sin 3.sin
4 2 4 2
π π
   
+ = −
 ÷  ÷
   
x x

2
2
x k
π
π
= +
2.sin(2x -
3
π
) = 5sin(x -
6
π
) + cos3x x =
6
π
+ k
π

3.2cos(
6
x
π
+
) = sin3x - cos3x
5
, ,
12 6 2
k k k
π π π
π π π

− + + − +
4.
2sinx 2sinx 1 2sin2x 2sin2x 1+ - = + -

x k2 ,k
3
p
= + p Î ¢
5
1
2cos 2 8cos 7
cos
x x
x
− + =

2 , 2
3
k k
π
π π
± +
,
6.
os2 5 2(2 cos )(sinx cos )C x x x+ = − −
2 , 2
2
k k
π
π π π

+ +
7.2sin
3
x – cos2x + cosx = 0
4
π
x nπ= − +
;
2x kπ=
8.
8 8
17
sin cos
32
x x+ =
8 4
π π
x k
= +
GOOD LUCK!!!!!