721 7.4 Double-Angle and Half-Angle Identities Find values of the sine and cosine functions for each angle measure. See Examples 1 and 2. 11. 2u, given sinu = 2 5 and cos u 60 12. 2u, given cos u = - 12 13 and sinu 70 13. 2x, given tanx = 2 and cos x 70 14. 2x, given tanx = 5 3 and sinx 60 15. 2u, given sinu = - 25 7 and cos u 70 16. 2u, given cos u = 23 5 and sinu 70 17. u, given cos 2u = 3 5 and u terminates in quadrant I 18. u, given cos 2u = 3 4 and u terminates in quadrant III 19. u, given cos 2u = - 5 12 and 90° 6u 6180° 20. u, given cos 2u = 2 3 and 90° 6u 6180° Simplify each expression. See Example 3. 21. cos2 15° - sin2 15° 22. 2 tan15° 1 - tan2 15° 23. 1 - 2 sin2 15° 24. 1 - 2 sin2 22 1 2 ° 25. 2 cos2 67 1 2 ° - 1 26. cos2 p 8 - 1 2 27. tan51° 1 - tan2 51° 28. tan34° 211 - tan2 34°2 29. 1 4 - 1 2 sin2 47.1° 30. 1 8 sin29.5° cos 29.5° 31. sin2 2p 5 - cos2 2p 5 32. cos2 2x - sin2 2x Express each function as a trigonometric function of x. See Example 4. 33. sin4x 34. cos 3x 35. tan3x 36. cos 4x Write each expression as a sum or difference of trigonometric functions. See Example 6. 37. 2 sin58° cos 102° 38. 2 cos 85° sin140° 39. 2 sin p 6 cos p 3 40. 5 cos 3x cos 2x 41. 6 sin4x sin5x 42. 8 sin7x sin9x Write each expression as a product of trigonometric functions. See Example 7. 43. cos 4x - cos 2x 44. cos 5x + cos 8x 45. sin25° + sin1-48°2 46. sin102° - sin95° 47. cos 4x + cos 8x 48. sin9x - sin3x Use a half-angle identity to find each exact value. See Examples 8 and 9. 49. sin67.5° 50. sin195° 51. tan195° 52. cos 195° 53. cos 165° 54. sin165° Use the given information to find each of the following. See Example 10. 55. cos x 2, given cos x = 1 4, with 0 6x 6 p 2 56. sin x 2, given cos x = - 5 8, with p 2 6x 6p 57. tan u 2, given sinu = 3 5, with 90° 6u 6180° 58. cos u 2, given sinu = - 4 5, with 180° 6u 6270° 59. sin x 2, given tanx = 2, with 0 6x 6 p 2 60. cos x 2, given cot x = -3, with p 2 6x 6p 61. tan u 2, given tanu = 27 3 , with 180° 6u 6270°
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