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Chapter 5 Practice Test
Name _________________
1. Graphically determine the general solution: 3sin(5x) = 1
A. x  0.07  2n , x  0.56  2n
B. x  0.07  2n , x  5.94  2n
5
5
5
5
C. x  0.07  2n , x  0.56  2n
D. x  0.07  2n , x  5.94  2n
2.
A.

3.
A.

4.
A.
Graph and solve: sin x +1 = sec x , 0  x  2 
0, 1.00
B. 0, 1.84
C. 1.00, 1.84
Solve:
0.52
D. 0, 1.00, 1.57, 4.71
2 sin x cos x  1, where 0  x  2. (Accurate to 2 dec. places.) 
B. 0.79
C. 0.52, 2.62
D. 0.79, 3.93
Determine the number of solutions for: (tanq + 1)(2sinq + 1)(cosq + 2) = 0, where 0  q < .
2
B. 3
C. 4
D. 6
2
5. How many solutions does sin x 
A. 1
6. Solve:
A.  , 2
3 3
B. 2
1
3
have over the interval 0  x  2 ?
C. 3
3  2 sin x  0, 0  x  2 (Give exact solutions) 
B. 4 , 5
C.  , 5
3 3
6 6
7. Solve: 4cos2x - 3 = 0 , 0  x  2
A.  , 5 , 7 and 11
6 6 6
6
C.  ,  , 3 , and 2
2
2
Give exact solution(s).
B.  , 2 , 4 and 5
3 3 3
3
D.  , 3 , 5 and 7
4 4 4
4
D. 4
D. 7 , 11
6 6
8. Solve : 1  2 sin x  0, 0  x  2 Give exact solution(s).
A. 
B. 2 , 5
C. 2 , 
D. 7 , 11
6 
3 3 
3 3 
6 6 




9. Determine the number of solutions in the interval 0 £ x < 2p for: sinax = 0.5,
a is an integer, where a  1
A. 2
B. a
2
C. a
10.
A.
B.
C.
D.
11.
A.
B.
C.
D.
D. 2a
12.
A.
B.
C.
D.
13. Which expression below is equivalent to 2 cot
A. 2 tan 5

B.
1
2 tan


5
?
2
C.
tan
5
D.
5

2
tan

5
1

2
B. 7  n, 11  n (n is any integer)

12
12
D. 13  n, 21  n (n is any integer)

12
12
14. Determine the general solution for: sin 2 x  
A. 7  2n, 11  2n (n is any integer)

12
12
C. 13  2n, 21  2n (n is any integer)

12
12







15. Sin( ) equals
A. –sin 
B. sin 




C. cos  
16. What is (are) the restriction(s) for the expression
A. sec x  0
B. sin x  0
D. –cos 
sec x
?
sin x
C. cos x  0
D. cos x  0, sin x  0

17. For what value of x is the following expression undefined?
A. 0
B. 

2
C. 
B. 0, 
C. 0, 
2
sin x
, where 0  x  2 
1  cos x
D. 3

2


18. Determine all values of x for which the following expression is undefined.
sin x
,
1 sec 2 x where 0  x  2 
A.  , 3
2 2
19. Simplify: 4 – 8 sin26x.
A. cos12x
B. 2 cos 6x






20. Determine an expression equivalent to
A.  csc 2
B. sec 2
C. 4 cos 6x
sec 2 
.
sec 2   2 
C. csc 2
D. 0,  ,  , 3
2 2
D. 4 cos 12x
D. sec 2
21. Which expression is equivalent to:
A. tan  tan2
B. 1  tan2
22. Simplify the expression:
A. 2
1  tan
?
1  cot
C. 1 – cot 

tan – 1
D. tan
C. sin
D. sin  cos
C. cos2q
D. cos4q
sin2

sin
B. 2 cos 
23. Simplify : cos4q - sin4q
A. –1
B. - cos2q
24.
A. sin x
B. -sin x
C. cos x
D. -cos x
25.
A.
B.
C.
D.
26. Which expression is equivalent to sin(180º + q)?
A. sin q
B. –sin q
C. cos q
D. –cos q
27. Which of the following is equivalent to cos5 cos -sin 5 sin  ? 
A. cos 4
B. cos6
C. cos 4  sin 4
D. cos 6  sin 6
28. Simplify: cos(p – 2x)
A. –cos 2x
B. –sin 2x
C. cos 2x
D. sin 2x
3
4
29. If cos A  5 , A in the fourth quadrant, and cos B  5 , B in the first quadrant, then
sin (A + B) equals
A.  7
25 












B. 0
C.
30. Which expression is equivalent to
A. cos2(4x)





2
31. Solve sin  
B. sin2(4x)
1
2
7
25

cos(8 x )  1
?
2
C. cos2(16x)
D. 1
D. sin2(16x)
for  to 2 decimal places where 0    2
32. Consider the equation:
a.
b. Give the general solution for this equation. (Solve over the set of real numbers giving exact
value solutions.)
33.
2
34. Solve cos x  cos x over the set of real numbers. Give exact value solutions.









35. Give all solutions to:
2cos3x +
36. Prove the following identity:
3=0
1+ cot
 cot 
1+ tan
over 0  x  2
(3 marks)
(3 marks)

Pr ove that 2 csc 2  
37.
1
1

1  cos  1 – cos 


38.
Prove the following identity.
sin   tan 
 tan 
1  cos 
(2 marks)

2
39. Prove the following identity: 2 – 4 sin   2 cos 2


40. Prove the following identity: csc 2 x 
1
sec x csc x
2

41. Prove algebraically the following identity:

sin 2 x
 cot x
1  cos 2 x







(2 marks)
42.
43.