1. Trigonometric Equations
Find the general solutions (express in radian) of following equations
2n
1. 2 cos(3x ) 1 0 Ans. n
6 3 12 18
3x 4n 2
2. 2 sin 1 0 Ans. ( 1) n 1
n
4 3 9
x
3. tan( ) 3 Ans. 4n n
4 3
4. sin x 1 cos x Ans. n ( 1) n n
4 4
5. 3 sin x cos x 3 Ans. n ( 1) n n
3 6
6. ( 3 1) cos x ( 3 1) sin x 2 Ans. 2n n
4 12
x 2
7. sin x sin Ans. 2n , 4n
2 3
n 2n
8. 3sin 7 x 2sin 4x 3sin x 0 Ans. , 0.4103
4 3
9. tan( ) tan( ) 4 Ans. n
4 4 6
(4n 1) (4n 1)
10. cos3 sin Ans. ,
8 4
Find the general solutions of the following equations and gives where .
n (2n 1) 2 3 3 4
11. sin 7 x sin3x 0 Ans. , ; 0, , , , , , ,
5 4 5 4 5 5 4 5
2n 2
12. sin x sin 2x sin3x Ans. , n ; 0, ,
3 3
n ( 1)n
3 4 ; 5 29
13. sin(2 x ) cos( x) Ans. n 1
, ,
4 6 2 ( 1) 12 36 36
n n 5 7 3 11
14. sin 4x cos 2 x 0 Ans. , ( 1)n 1
; , , , , ,
2 4 2 12 12 12 4 4 12 4 12
15. 3sin 2x 2tan x Ans. n , n 0.9553 ; 0,0.9553, 2.1866,

2. Find the general solutions of the following equations.
3
16. 2sin3x cos 2x 1 Ans. n , n ( 1)n 0.8481, n ( 1) n
2
17. 5cos 2x 8sin x 3 Ans. n ( 1) n 0.2014 , 2n
2
2n 2n n
18. sin5x sin3x sin8x Ans. , ,
5 10 3 6 4
19. 2 cos 2 3 sin 1 0 Ans. n ( 1)n 1
3
3 n
20. cos3 x sin x sin 3 x cos x Ans. ( 1) n
8 4 12
21. 4cos 3sin 3 Ans. 2n 0.9273 0.6435
22. cos 2 x 4sin 2 x cos x 2 0 Ans. (2n 1) , 2n
3
n (2n 1)
23. sin 2 sin 2 2 sin 2 3 Ans. ,
2 6
24. a) Prove that cos3 4cos3 3cos .
b) Solve the equation cos 3 cos ,where 0
6
11 13
Ans. , ,
18 18 18
c) By the results of questions a) and b) ,deduce the roots of the equation
3
4 x3 3x .Correct the answer to 3 decimal places.
2
Ans. 0.985, 0.342, 0.643