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Exam 2 Study Guide Math 123: College Trigonometry March 31, 2016 1. Verify the trigonometric identity: cos2 (x) − sin2 (x) = 2 cos2 (x) − 1 2. Verify the trigonometric identity: sin2 (x) − sin4 (x) = cos2 (x) − cos4 (x) 3. Verify the trigonometric identity: cos(x)(tan2 (x) + 1) = sec(x) 4. Verify the trigonometric identity: 1+sin(x) cos(x) + cos(x) 1+sin(x) = 2 sec(x) 5. Verify the trigonometric identity: 1 cos(x)+1 + 1 cos(x)−1 = −2 csc(x) cot(x) 6. Verify the trigonometric identity: tan5 (x) = tan3 (x) sec2 (x) − tan3 (x) 7. Verify the trigonometric identity: (sin(x) + cos(x))2 = 1 + sin(2x) 8. Solve: sin(x) = √ 3 − sin(x) in the interval [0, 2π). 9. Solve: 4 cos(x) = 1 + 2 cos(x) in the interval [0, 2π). 10. Solve: 4 tan2 (x) − 1 = tan2 (x) in the interval [0, 2π). 11. Solve: 2 cos2 (x) − cos(x) = 1 in the interval [0, 2π). 12. Solve: 2 cos2 (x) + 3 cos(x) = 0 in the interval [0, 2π). 13. Solve: 2 sin2 (x) + 3 sin(x) + 1 = 0 in the interval [0, 2π). 14. Solve: 2 sin2 (x) = 2 + cos(x) in the interval [0, 2π). 15. Solve: cos2 (x) + sin(x) = 1 in the interval [0, 2π). 16. Solve: sin2 (x) + 2 cos(x) = 2 in the interval [0, 2π). 17. Solve: cos(x) + 1 = sin(x) in the interval [0, 2π). 18. Solve: sec(x) + tan(x) = 1 in the interval [0, 2π). 19. Solve: 2 cos(2x) − 1 = 0 in the interval [0, 2π). 20. Solve: 2 sin(2x) + √ 3 = 0 in the interval [0, 2π). 21. Given t = a) sin(t) b) cos(t) 7π , 12 find the exact values of:(Hint: 7π = 12 22. Given t = a) sin(t) b) cos(t) −π , 12 find the exact values of:(Hint: −π = 12 23. Given sin(x) = a) sin(x + y) b) cos(x − y) c) tan(x + y) 5 13 π 3 + π4 ) π 6 − π4 ) and cos(y) = − 35 with both x and y are in Quadrant II. Find the exact value for: 7 24. Given sin(x) = − 25 and cos(y) = − 45 with both x and y are in Quadrant III. Find the exact value for: a) sin(x + y) b) cos(x + y) c) tan(x − y) 25. Solve: cos(x + π) − cos(x) − 1 = 0 in the interval [0, 2π). 26. Given that sin(x) = − 35 for a) sin(2x) b) cos(2x) c) tan(2x) 3π 2 27. Given that cos(x) = − 45 for a) sin(2x) b) cos(2x) c) tan(2x) π 2 < x < 2π. Find the exact values for: < x < π. Find the exact values for: 28. Given x = π8 , use the half angle formulas to determine the exact values for: a) sin(x) b) cos(x) c) tan(x) 29. Solve: sin( x2 ) + cos(x) = 0 in the interval [0, 2π). 30. Solve: sin( x2 ) + cos(x) − 1 = 0 in the interval [0, 2π). 31. Use the sum-to-product formula to find the exact value of: cos 75◦ + cos 15◦ 32. Use the sum-to-product formula to find the exact value of: cos 75◦ − cos 15◦ 33. Use the product-to-sum formulas to rewrite the product 2 sin(7x) cos(3x) as a sum or difference. 34. Use the product-to-sum formulas to rewrite the product cos(4x) sin(6x) as a sum or difference.