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Mesa College – Math 104 ( Trig) Challenge Exam SAMPLES Directions: NO CALCULATOR. Write neatly and show your work and steps. Answers without appropriate work shown will receive little or NO credit. Be sure to simplify all radicals and fractions. Attach your neat and organized solution sheets behind this cover sheet. Make sure each solution is properly labeled. #1 – 3. Use the given information to find the values of the remaining Trigonometric functions. 1. If sin θ = − cos θ = 1 and 2 2. If sin θ = 3. If cot θ = −4 and 3 and θ is in QII, 5 270o ≤ θ < 360o 3 2 #4-6 . Find ALL values of x. Express your answers using radians. 4. 3csc x − 6 = 0. 5. tan 2 x = − tan x 6. 2 sin 2 x − 5sin x − 3 = 0 7. Find the side length x in each of the triangles shown. (7b) (7a) 10 60 ° 40 ° x 8. If cos x = 0.8 and sin x = −0.6 , find : (8a) sin(2 x ) 9. Given that cos α = (8b) cos(2 x ) (8c) in which quadrant is angle (2x) ? 2 1 −π π and sin β = − , where ≤ β ≤ and 0 ≤ α ≤ π find : 3 4 2 2 (9a) sin (α + β ) α 2 (9b) cos(α − β ) (9c) cos (9d) in which quadrant is angle ( α + β )? (9e) in which quadrant is angle ( α − β )? α ? 2 (9f) in which quadrant is angle 10. Find the principal value of each: − 3 2 10a) Arc cos 10b) sin −1 − 1 2 10c) 3 tan −1 3 10d) arc cot ( 3) 10e) cos arcsin 3 2 10f) cos ( arctan(−1) ) 10g) cot −1 sin 3π 2 11. Find the exact values of each: 11a) π sin 4 −3π 4 ( ) 13π 6 11e) cot ( 11d) sec −420o ) 11b) cos 30o 11c) csc π sin 6 11f) π 1 + cos 6 11g) cos 45o cos 60o − sin 45o sin 60o ( ) ( ) ( ) ( ) 12. Simplify each expression completely. o 3π 2 3π − cos 8 8 5π 8 (12b) sin 2 o (12a) sin(75 ) cos(15 ) − cos(75) sin(15) (12c) cos 13. Find all solutions between 0 ≤ θ < 360o for each: x 2 (13a) cos 2 θ − 2sin θ = −2 (13c) sin(2 x) = cos( x) (13b) csc = 2 14. Sketch at least one period of the graph of each. Label the coordinates of one maximum point and the coordinates of one minimum point. (14a) f ( x) = −2 cos x + 15. Simplify the expression (a) cos y π +1 2 (14b) g ( x) = 5sin 2 x − π (14c) h( x) = 3 + 2sec x 4 cot y − 1 , so that it matches one of the expressions below. SHOW STEPS. 1 − tan y (b) tan y (c ) 0 (d) sec y csc y (e) cot y 16. Simplify each expression. (16a) (sec β − tan β )(sec β + tan β ) (16b) cos θ 1 + sin θ + 1 + sin θ cos θ 17. Complete the trigonometric identity: sec θ − cos θ = w , by selecting w from the list of expressions below. (a) tan θ sin θ (b) cot 2 θ cos θ (c ) 1 − cos θ sin θ sin θ (d) sec θ − 1 sec θ 18. SIMPLIFY each to a single trig expression involving no fractions. (18a) 1 + tan 2 x 1 + tan 2 x (18b) ( sin ( ) + cos ( ) ) x 2 x 2 2 −1 (e) -1 19. If A = 3 + 2i and B = 5 − i , where i = −1 , find each: (19a) A2 (19b) AB 20. Evaluate: 2i 2 − 5i 75 + (19c) A B 6 , where i = −1 . i13 21. Express each in standard form (i.e. a ± bi ) (21a) 5 2 cos ( ) − i sin 5π 5π 4 4 (21b) 8cis30o 22. Convert each from Polar coordinates to Cartesian coordinates (22a) ( 2, 30 ) o (22b) −6, 4π 3 23. Convert each from Cartesian coordinates to Polar coordinates ( (23a) 3, 3 ) (23b) ( −5, −5) 4 24. If 4 + 3i ≈ 5cis37 o , find ( 4 + 3i ) in trigonometric form. ********************************************************************************************** Answers to Math 104 (Trig) SAMPLES { This is a DRAFT… If you find any errors, please contact me at: [email protected] … Thanks. } 1. sin θ = −1 3 2 3 − 3 , cos θ = , csc θ = −2 , sec θ = , tan θ = , cot θ = − 3 2 2 3 3 2. sin θ = 3 −4 5 −5 −3 −4 , cos θ = , csc θ = , sec θ = , tan θ = , cot θ = 5 5 3 4 4 3 3. sin θ = − 17 4 17 17 −1 , cos θ = , csc θ = − 17 , sec θ = , tan θ = , cot θ = −4 17 17 4 4 π 7π 11π + 2kπ or x = + 2k π 6 6 4. x | x = 6. x | x = 6 + 2kπ or x = 5π + 2 kπ 6 5. x | x = kπ or x = 7a. x = 10 39 3π + kπ 4 7b. x = 20sin 80o 3 8a. −24 = − 0.96 25 8b. 9c. + 30 6 9d. 9e. 9f. all are in Q-I 10e. 1 2 10f. 11c. − 2 13b. 2 2 2 2 12b. {90 , 270 } o o 15. E 16a. 1 18a. cos 2 x π 5π 4 23b. 5 2, 12c. 2− 2 2 { 6 24. 625cis (148o ) 5 3−2 12 5π 6 10b. 9b. −π 6 2 2 11a. 10c. { } 14. GRAPHS… see below 16b. 2sec θ 17. A 19a. 5 + 12i 1 1 + i 2 2 21b. 4 3 + 4i 20. −2 − i 22a. ( ) 3,1 ( 22b. 3, 3 3 π 6 2− 6 4 11g. 13a. 90o } 2 15 − 5 12 3 2 11b. 11f. 2 − 3 3 13c. 90o , 270o , 30o ,150o 19c. 21a. −5 + 5i 9a. 10a. 11e. 18b. sin x 19b. 17 + 7i 23a. 2 3, 8c. Q-IV 10g. undefined. 11d. 2 3 2 12a. 7 = 0.28 25 ) 10d. π 6