749 7.6 Trigonometric Equations 33. csc2 u - 2 cot u = 0 34. sin2 u cos u = cos u 35. 2 tan u sin u - tan u = 0 36. sin u cos u = 0 37. sec2 u tan u = 2 tan u 38. cos2 u - sin2 u = 0 39. 9 sin2 u - 6 sin u = 1 40. 4 cos2 u + 4 cos u = 1 41. tan2 u + 4 tan u + 2 = 0 42. 3 cot2 u - 3 cot u - 1 = 0 43. sin2 u - 2 sin u + 3 = 0 44. 2 cos2 u + 2 cos u + 1 = 0 45. cot u + 2 csc u = 3 46. 2 sin u = 1 - 2 cos u Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1–5. 47. cos u + 1 = 0 48. tan u + 1 = 0 49. 3 csc x - 223 = 0 50. cot x + 23 = 0 51. 6 sin2 u + sin u = 1 52. 3 sin2 u - sin u = 2 53. 2 cos2 x + cos x - 1 = 0 54. 4 cos2 x - 1 = 0 55. sin u cos u - sin u = 0 56. tan u csc u - 23 csc u = 0 57. sin x13 sin x - 12 = 1 58. tan x1tan x - 22 = 5 59. 5 + 5 tan2 u = 6 sec u 60. sec2 u = 2 tan u + 4 61. 2 tan u 3 - tan2 u = 1 62. 2 cot2 u cot u + 3 = 1 The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval 30, 2p2. Express solutions to four decimal places. 63. x2 + sin x - x3 - cos x = 0 64. x3 - cos2 x = 1 2 x - 1 Concept Check Answer each question. 65. Suppose solving a trigonometric equation for solutions over the interval 30, 2p2 leads to 2x = 2p 3 , 2p, 8p 3 . What are the corresponding values of x? 66. Suppose solving a trigonometric equation for solutions over the interval 30, 2p2 leads to 1 2x = p 16, 5p 12 , 5p 8 . What are the corresponding values of x? 67. Suppose solving a trigonometric equation for solutions over the interval 30°, 360°2 leads to 3u = 180°, 630°, 720°, 930°. What are the corresponding values of u? 68. Suppose solving a trigonometric equation for solutions over the interval 30°, 360°2 leads to 1 3u = 45°, 60°, 75°, 90°. What are the corresponding values of u? Solve each equation in x over the interval 30, 2p2 and each equation in u over the interval 30°, 360°2. Give exact solutions. See Examples 6–8. 69. 2 cos 2x = 23 70. 2 cos 2x = -1 71. sin 3u = -1 72. sin 3u = 0 73. 3 tan 3x = 23 74. cot 3x = 23 75. 22 cos 2u = -1 76. 223 sin 2u = 23 77. sin x 2 = 22 - sin x 2 78. tan 4x = 0 79. sin x = sin 2x 80. cos 2x - cos x = 0
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