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Trigonometric Identities
Exam Questions
Name: ANSWERS
2 January 2013 – January 2017
3 January 2013 – January 2017
Multiple Choice
1. Simplify the following expression:
xx 22 cot1cos
a. x2sin b. x2cos c. x2cot d. x2sec
2. Identify a non-permissible value of x for the expression x2cos
1.
a. 0 b. 4
c.
2
d.
3. Evaluate: 8
cos8
sin2
a. 2
1 b.
2
2 c. 1 d. 2
4 January 2013 – January 2017
4. A non-permissible value of x for the function 1cos
1
xxf is:
a. -1 b. 0 c. d. 2
3
5. Identify the trigonometric function that is equivalent to 3
sin4
cos3
cos4
sin
.
a. 7
2sin
b.
12
7sin
c.
7
2cos
d.
12
7cos
5 January 2013 – January 2017
Written Response
6. On the interval 20 , identify the non-permissible values of for the trigonometric
identity:
cot
1tan (2 marks)
7. Explain the error that was made when solving the following equation: (1 mark) R where,cos2sin
6 January 2013 – January 2017
8. The graph of xy 2sin is sketched below.
Explain how to use this graph to solve the equation 2
12sin x over the interval 2 ,0 .
(1 mark)
9. Determine all non-permissible values of over the interval [0, 2 ]
cotcsc
cos1
sin
Explain your reasoning. (3 marks)
7 January 2013 – January 2017
10. Determine the exact value of: (3 marks)
12
11cos4
8 January 2013 – January 2017
11. Prove the identity below for all permissible values of x : (3 marks)
xx
xcot
2sin
2cos1
9 January 2013 – January 2017
12. Prove the identity below for all permissible values of x : (3 marks)
xxx
x 22
coscos1sec
sin
10 January 2013 – January 2017
13. Given an example using the values for A and B , in degrees or radians, to verify that
BABA coscoscos is not an identity. (2 marks)
11 January 2013 – January 2017
14. Solve the following equation algebraically where 00 360180 . (calculator)
01cos5sin2 2 (4 marks)
15. Find the exact value of
12
19sin
. (3 marks)
12 January 2013 – January 2017
16. Solve the following equation over the interval 2 ,0 . (4 marks)
012cos2
13 January 2013 – January 2017
17. a. Prove the identity below for all permissible values of . (2 marks)
3tancos
cos21 2
2
2
b. Determine all the non-permissible values of . (2 marks)
14 January 2013 – January 2017
18. Given that 13
5sin , where is in Quadrant II, and
5
2cos , where is in Quadrant IV,
find the exact value of:
a. cos (3 marks)
b. 2sin (1 mark)
15 January 2013 – January 2017
19. Prove the identity for all permissible values of : (3 marks)
2cos
tan1
tan12
2
16 January 2013 – January 2017
20. Solve the following equation algebraically for x , where 20 x .
xx sin3cos2 2 (4 marks)
21. Given 5
3cos , where is in quadrant IV, and
3
2cos , where is in quadrant II,
determine the exact value of sin . (3 marks)
17 January 2013 – January 2017
22. Prove the identity below for all permissible values of . (3 marks)
sin
cotcsc
cos1
1 2
18 January 2013 – January 2017
23. a. Verify that the equation x
x
x
x
sin2
2sin
cos
sin1 2
is true for 3
x . (2 marks)
b. Explain why verifying the equation for 3
x is insufficient to conclude that the equation is
an identity. (1 mark)
19 January 2013 – January 2017
24. Solve the following equation algebraically over the interval 2,0 .
02sin32cos (4 marks)
25. Over the interval 2,0 , determine the non-permissible values of in the expression
1coscsc . (2 marks)
20 January 2013 – January 2017
26. Determine the exact value of 12
13sin
. (3 marks)
27. Given that 5
2cot , where is in Quadrant IV, determine the exact value of sin .
(2 marks)
21 January 2013 – January 2017
28. Prove the identity for all permissible values of x . (3 marks)
x
xxx
sin1
costansec
22 January 2013 – January 2017
29. Prove the identity below for all permissible values of : (3 marks)
tancos
1
tan
cossin
Solution
LHS RHS
sin
1
sin
cossin
sin
cos
sin
sin
sin
cossin
sin
coscossin
cos
sin
cossin
22
22
2
sin
1
cos
sincos
1
tancos
1
23 January 2013 – January 2017
30. Determine the exact value of o75tan . (2 marks)
Solution:
13
31
13
3
3
31
3
13
3
31
13
11
13
1
45tan30tan1
45tan30tan
4530tan75tan
oo
oo
ooo
31. Solve the following equation algebraically for , where 20 :
12cos2 (4 marks)
24 January 2013 – January 2017
32. Prove the identity for all permissible values of : (3 marks)
2cos1
sintansintancos
25 January 2013 – January 2017
33. Given that 12
7cos where is in quadrant IV, and
5
3sin where is in quadrant I,
determine the exact value of:
a. sin (3 marks)
b. csc (1 mark)
26 January 2013 – January 2017
34. Given 3
1cot , where is in quadrant II, determine the exact value of sin .
(2 marks)
27 January 2013 – January 2017
35. Given the identity
cos
sin2cossec
2 .
a. Determine the non-permissible values of , over the interval 20 .
(1 mark)
b. prove the identity for all permissible values of . (3 marks)
28 January 2013 – January 2017
36. Given that 7
3sin , where is in Quadrant II, and
5
4cos , where is in Quadrant IV,
determine the exact value of:
a. sin (3 marks)
b. 2cos
