Solutions for Solving Trigonometric Equations

Solutions to Try Its

1. [latex]x=\frac{7\pi }{6},\frac{11\pi }{6}[/latex]

2. [latex]\frac{\pi }{3}\pm \pi k[/latex]

3. [latex]\theta \approx 1.7722\pm 2\pi k[/latex] and [latex]\theta \approx 4.5110\pm 2\pi k[/latex]

4. [latex]\cos \theta =-1,\theta =\pi [/latex]

5. [latex]\frac{\pi }{2},\frac{2\pi }{3},\frac{4\pi }{3},\frac{3\pi }{2}[/latex]

Solutions to Odd-Numbered Exercises

1. There will not always be solutions to trigonometric function equations. For a basic example, [latex]\cos \left(x\right)=-5[/latex].

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. [latex]\frac{\pi }{3},\frac{2\pi }{3}[/latex]

7. [latex]\frac{3\pi }{4},\frac{5\pi }{4}[/latex]

9. [latex]\frac{\pi }{4},\frac{5\pi }{4}[/latex]

11. [latex]\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}[/latex]

13. [latex]\frac{\pi }{4},\frac{7\pi }{4}[/latex]

15. [latex]\frac{7\pi }{6},\frac{11\pi }{6}[/latex]

17. [latex]\frac{\pi }{18},\frac{5\pi }{18},\frac{13\pi }{18},\frac{17\pi }{18},\frac{25\pi }{18},\frac{29\pi }{18}[/latex]

19. [latex]\frac{3\pi }{12},\frac{5\pi }{12},\frac{11\pi }{12},\frac{13\pi }{12},\frac{19\pi }{12},\frac{21\pi }{12}[/latex]

21. [latex]\frac{1}{6},\frac{5}{6},\frac{13}{6},\frac{17}{6},\frac{25}{6},\frac{29}{6},\frac{37}{6}[/latex]

23. [latex]0,\frac{\pi }{3},\pi ,\frac{5\pi }{3}[/latex]

25. [latex]\frac{\pi }{3},\pi ,\frac{5\pi }{3}[/latex]

27. [latex]\frac{\pi }{3},\frac{3\pi }{2},\frac{5\pi }{3}[/latex]

29. [latex]0,\pi [/latex]

31. [latex]\pi -{\sin }^{-1}\left(-\frac{1}{4}\right),\frac{7\pi }{6},\frac{11\pi }{6},2\pi +{\sin }^{-1}\left(-\frac{1}{4}\right)[/latex]

33. [latex]\frac{1}{3}\left({\sin }^{-1}\left(\frac{9}{10}\right)\right),\frac{\pi }{3}-\frac{1}{3}\left({\sin }^{-1}\left(\frac{9}{10}\right)\right),\frac{2\pi }{3}+\frac{1}{3}\left({\sin }^{-1}\left(\frac{9}{10}\right)\right),\pi -\frac{1}{3}\left({\sin }^{-1}\left(\frac{9}{10}\right)\right),\frac{4\pi }{3}+\frac{1}{3}\left({\sin }^{-1}\left(\frac{9}{10}\right)\right),\frac{5\pi }{3}-\frac{1}{3}\left({\sin }^{-1}\left(\frac{9}{10}\right)\right)[/latex]

35. [latex]0[/latex]

37. [latex]\frac{\pi }{6},\frac{5\pi }{6},\frac{7\pi }{6},\frac{11\pi }{6}[/latex]

39. [latex]\frac{3\pi }{2},\frac{\pi }{6},\frac{5\pi }{6}[/latex]

41. [latex]0,\frac{\pi }{3},\pi ,\frac{4\pi }{3}[/latex]

43. There are no solutions.

45. [latex]{\cos }^{-1}\left(\frac{1}{3}\left(1-\sqrt{7}\right)\right),2\pi -{\cos }^{-1}\left(\frac{1}{3}\left(1-\sqrt{7}\right)\right)[/latex]

47. [latex]{\tan }^{-1}\left(\frac{1}{2}\left(\sqrt{29}-5\right)\right),\pi +{\tan }^{-1}\left(\frac{1}{2}\left(-\sqrt{29}-5\right)\right),\pi +{\tan }^{-1}\left(\frac{1}{2}\left(\sqrt{29}-5\right)\right),2\pi +{\tan }^{-1}\left(\frac{1}{2}\left(-\sqrt{29}-5\right)\right)[/latex]

49. There are no solutions.

51. There are no solutions.

53. [latex]0,\frac{2\pi }{3},\frac{4\pi }{3}[/latex]

55. [latex]\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}[/latex]

57. [latex]{\sin }^{-1}\left(\frac{3}{5}\right),\frac{\pi }{2},\pi -{\sin }^{-1}\left(\frac{3}{5}\right),\frac{3\pi }{2}[/latex]

59. [latex]{\cos }^{-1}\left(-\frac{1}{4}\right),2\pi -{\cos }^{-1}\left(-\frac{1}{4}\right)[/latex]

61. [latex]\frac{\pi }{3},{\cos }^{-1}\left(-\frac{3}{4}\right),2\pi -{\cos }^{-1}\left(-\frac{3}{4}\right),\frac{5\pi }{3}[/latex]

63. [latex]{\cos }^{-1}\left(\frac{3}{4}\right),{\cos }^{-1}\left(-\frac{2}{3}\right),2\pi -{\cos }^{-1}\left(-\frac{2}{3}\right),2\pi -{\cos }^{-1}\left(\frac{3}{4}\right)[/latex]

65. [latex]0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}[/latex]

67. [latex]\frac{\pi }{3},{\cos }^{-1}\left(-\frac{1}{4}\right),2\pi -{\cos }^{-1}\left(-\frac{1}{4}\right),\frac{5\pi }{3}[/latex]

69. There are no solutions.

71. [latex]\pi +{\tan }^{-1}\left(-2\right),\pi +{\tan }^{-1}\left(-\frac{3}{2}\right),2\pi +{\tan }^{-1}\left(-2\right),2\pi +{\tan }^{-1}\left(-\frac{3}{2}\right)[/latex]

73. [latex]2\pi k+0.2734,2\pi k+2.8682[/latex]

75. [latex]\pi k - 0.3277[/latex]

77. [latex]0.6694,1.8287,3.8110,4.9703[/latex]

79. [latex]1.0472,3.1416,5.2360[/latex]

81. [latex]0.5326,1.7648,3.6742,4.9064[/latex]

83. [latex]{\sin }^{-1}\left(\frac{1}{4}\right),\pi -{\sin }^{-1}\left(\frac{1}{4}\right),\frac{3\pi }{2}[/latex]

85. [latex]\frac{\pi }{2},\frac{3\pi }{2}[/latex]

87. There are no solutions.

89. [latex]0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}[/latex]

91. There are no solutions.

93. [latex]{7.2}^{\circ }[/latex]

95. [latex]{5.7}^{\circ }[/latex]

97. [latex]{82.4}^{\circ }[/latex]

99. [latex]{31.0}^{\circ }[/latex]

101. [latex]{88.7}^{\circ }[/latex]

103. [latex]{59.0}^{\circ }[/latex]

105. [latex]{36.9}^{\circ }[/latex]