1. There will not always be solutions to trigonometric function equations. For a basic example, cos(x)=−5.
3. If the sine or cosine function has a coefficient of one, isolate the term on one side of the equals sign. If the number it is set equal to has an absolute value less than or equal to one, the equation has solutions, otherwise it does not. If the sine or cosine does not have a coefficient equal to one, still isolate the term but then divide both sides of the equation by the leading coefficient. Then, if the number it is set equal to has an absolute value greater than one, the equation has no solution.
5. 3π,32π
7. 43π,45π
9. 4π,45π
11. 4π,43π,45π,47π
13. 4π,47π
15. 67π,611π
17. 18π,185π,1813π,1817π,1825π,1829π
19. 123π,125π,1211π,1213π,1219π,1221π
21. 61,65,613,617,625,629,637
23. 0,3π,π,35π
25. 3π,π,35π
27. 3π,23π,35π
29. 0,π
31. π−sin−1(−41),67π,611π,2π+sin−1(−41)
33. 31(sin−1(109)),3π−31(sin−1(109)),32π+31(sin−1(109)),π−31(sin−1(109)),34π+31(sin−1(109)),35π−31(sin−1(109))
35. 0
37. 6π,65π,67π,611π
39. 23π,6π,65π
41. 0,3π,π,34π
43. There are no solutions.
45. cos−1(31(1−7)),2π−cos−1(31(1−7))
47. tan−1(21(29−5)),π+tan−1(21(−29−5)),π+tan−1(21(29−5)),2π+tan−1(21(−29−5))
49. There are no solutions.
51. There are no solutions.
53. 0,32π,34π
55. 4π,43π,45π,47π
57. sin−1(53),2π,π−sin−1(53),23π
59. cos−1(−41),2π−cos−1(−41)
61. 3π,cos−1(−43),2π−cos−1(−43),35π
63. cos−1(43),cos−1(−32),2π−cos−1(−32),2π−cos−1(43)
65. 0,2π,π,23π
67. 3π,cos−1(−41),2π−cos−1(−41),35π
69. There are no solutions.
71. π+tan−1(−2),π+tan−1(−23),2π+tan−1(−2),2π+tan−1(−23)
73. 2πk+0.2734,2πk+2.8682
75. πk−0.3277
77. 0.6694,1.8287,3.8110,4.9703
79. 1.0472,3.1416,5.2360
81. 0.5326,1.7648,3.6742,4.9064
83. sin−1(41),π−sin−1(41),23π
85. 2π,23π
87. There are no solutions.
89. 0,2π,π,23π
91. There are no solutions.
93. 7.2∘
95. 5.7∘
97. 82.4∘
99. 31.0∘
101. 88.7∘
103. 59.0∘
105. 36.9∘