Trigonometric Identities Exam Questions Name: ANSWERS
January 01 January 017
Multiple Choice 1. Simplify the following expression: cos x 1 cot x a. sin x b. cos x c. cot x d. sec x. Identify a non-permissible value of x for the expression 1 cos x. a. 0 b. 4 c. d.. Evaluate: sin cos 8 8 a. 1 b. c. 1 d. January 01 January 017
4. A non-permissible value of x for the function x 1 f is: cos x 1 a. -1 b. 0 c. d. 5. Identify the trigonometric function that is equivalent to sin cos cos sin. 4 4 a. sin b. 7 7 sin c. 1 cos d. 7 7 cos 1 January 01 January 017 4
Written Response 6. On the interval 0, identify the non-permissible values of for the trigonometric 1 identity: tan ( marks) cot 7. Explain the error that was made when solving the following equation: (1 mark) sin cos, where R January 01 January 017 5
8. The graph of y sin x is sketched below. Explain how to use this graph to solve the equation sin 1 x over the interval, 0. (1 mark) 9. Determine all non-permissible values of over the interval [0, ] sin csc cot 1 cos Explain your reasoning. ( marks) January 01 January 017 6
10. Determine the exact value of: ( marks) 11 4cos 1 January 01 January 017 7
11. Prove the identity below for all permissible values of x : ( marks) 1 cos x cot x sin x January 01 January 017 8
1. Prove the identity below for all permissible values of x : ( marks) sin x cos x cos x sec x 1 January 01 January 017 9
1. Given an example using the values for A and B, in degrees or radians, to verify that cos A B cos A cos is not an identity. ( marks) B January 01 January 017 10
0 14. Solve the following equation algebraically where 180 0 60. (calculator) sin 5cos 1 0 (4 marks) 19 15. Find the exact value of sin. ( marks) 1 January 01 January 017 11
16. Solve the following equation over the interval, cos 1 0 0. (4 marks) January 01 January 017 1
17. a. Prove the identity below for all permissible values of. ( marks) 1 cos tan cos b. Determine all the non-permissible values of. ( marks) January 01 January 017 1
5 18. Given that sin, where is in Quadrant II, and 1 find the exact value of: cos, where is in Quadrant IV, 5 a. cos ( marks) b. sin (1 mark) January 01 January 017 14
19. Prove the identity for all permissible values of : ( marks) 1 tan cos 1 tan January 01 January 017 15
0. Solve the following equation algebraically for x, where 0 x. cos x sin x (4 marks) 1. Given cos, where is in quadrant IV, and cos, where is in quadrant II, 5 sin. ( marks) determine the exact value of January 01 January 017 16
. Prove the identity below for all permissible values of. ( marks) 1 cot csc 1 cos sin January 01 January 017 17
1 sin x. a. Verify that the equation cos x sin x sin x is true for x. ( marks) b. Explain why verifying the equation for an identity. x is insufficient to conclude that the equation is (1 mark) January 01 January 017 18
4. Solve the following equation algebraically over the interval 0,. cos sin 0 (4 marks) 5. Over the interval 0,, determine the non-permissible values of in the expression csc cos 1. ( marks) January 01 January 017 19
1 6. Determine the exact value of sin. 1 ( marks) 7. Given that cot, where is in Quadrant IV, determine the exact value of sin. 5 ( marks) January 01 January 017 0
8. Prove the identity for all permissible values of x. ( marks) cos x sec x tan x 1 sin x January 01 January 017 1
9. Prove the identity below for all permissible values of : ( marks) cos 1 sin tan cos tan Solution LHS cos sin sin cos cos sin cos sin cos sin sin sin cos sin sin sin cos sin 1 sin RHS 1 cos tan 1 sin cos cos 1 sin January 01 January 017
0. Determine the exact value of Solution: tan 75 o 1 tan 0 o o tan0 tan 45 o o 1 tan0 tan 45 1 1 1 1 1 1 o 1 1 1 45 o 1 o tan 75. ( marks) 1. Solve the following equation algebraically for, where 0 : cos 1 (4 marks) January 01 January 017
. Prove the identity for all permissible values of : ( marks) tan sin cos tan sin 1 cos January 01 January 017 4
7. Given that cos where is in quadrant IV, and sin where is in quadrant I, 1 5 determine the exact value of: a. sin ( marks) b. csc (1 mark) January 01 January 017 5
4. Given 1 cot, where is in quadrant II, determine the exact value of sin. ( marks) January 01 January 017 6
sin 5. Given the identity sec cos. cos a. Determine the non-permissible values of, over the interval 0. (1 mark) b. prove the identity for all permissible values of. ( marks) January 01 January 017 7
4 6. Given that sin, where is in Quadrant II, and cos, where is in Quadrant IV, 7 5 determine the exact value of: a. sin ( marks) b. cos January 01 January 017 8