SECTION 7.3 Trigonometric Equations 499 10. True or False The equation θ = sin 2 has a real solution that can be found using a calculator. 9. True or False The solution set of the equation θ = tan 1 is given by θ θ π π { } = +k k 4 , aninteger. 11. Multiple Choice If all solutions of a trigonometric equation are given by the general formula θ π π = + k 6 2 or θ π π = + k 11 6 2 , where k is an integer, then which of the following is not a solution of the equation? (a) π 35 6 (b) π 23 6 (c) π 13 6 (d) π7 6 12. Multiple Choice Suppose θ π = 2 is the only solution of a trigonometric equation in the interval θ π ≤ < 0 2 . Assuming a period of π2 , which of the following formulas gives all solutions of the equation, where k is an integer? (a) θ π π = + k 2 2 (b) θ π π = +k 2 (c) θ π = k 2 (d) θ π π = +k 2 Skill Building In Problems 13–36, solve each equation on the interval θ π ≤ < 0 2 . 13. θ + = 2 sin 3 2 14. θ − = 1 cos 1 2 15. θ + = 2 sin 1 0 16. θ + = cos 1 0 17. θ + = tan 1 0 18. θ + = 3 cot 1 0 19. θ + = − 4 sec 6 2 20. θ − = 5csc 3 2 21. θ + = − 3 2 cos 2 1 22. θ + = 4 sin 3 3 3 23. θ = 4cos 1 2 24. θ = tan 1 3 2 25. θ − = 2 sin 1 0 2 26. θ − = 4cos 3 0 2 27. θ ( ) = − sin 3 1 28. θ = tan 2 3 29. θ ( ) = − cos 2 1 2 30. θ ( ) = − tan 2 1 31. θ = − sec 3 2 2 32. θ = − cot 2 3 3 33. θ π ( ) − = − cos 2 2 1 34. θ π ( ) + = sin 3 18 1 35. θ π ( ) + = tan 2 3 1 36. θ π ( ) − = cos 3 4 1 2 In Problems 37–48, solve each equation. Give a general formula for all the solutions. List six solutions. 37. θ = sin 1 2 38. θ = tan 1 39. θ = − tan 3 3 40. θ = − cos 3 2 41. θ = cos 0 42. θ = sin 2 2 43. θ − = 3 cot 0 44. θ − = 2 3 csc 0 45. θ ( ) = − cos 2 1 2 46. θ ( ) = − sin 2 1 47. θ = − sin 2 3 2 48. θ = − tan 2 1 In Problems 49–60, use a calculator to solve each equation on the interval θ π ≤ < 0 2 . Round answers to two decimal places. 49. θ = sin 0.4 50. θ = cos 0.6 51. θ = tan 5 52. θ = cot 2 53. θ = − cos 0.9 54. θ = − sin 0.2 55. θ = − sec 4 56. θ = − csc 3 57. θ + = 5 tan 9 0 58. θ = − 4 cot 5 59. θ − = 3sin 2 0 60. θ + = 4cos 3 0 In Problems 61–84, solve each equation on the interval θ π ≤ < 0 2 . 61. θ θ + = 2cos cos 0 2 62. θ − = sin 1 0 2 63. θ θ − − = 2 sin sin 1 0 2 64. θ θ + − = 2cos cos 1 0 2 65. θ θ ( )( ) − − = tan 1 sec 1 0 66. θ θ ( ) ( ) + − = cot 1 csc 1 2 0
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