10.7E: Exercises

Exercises

In Exercises 1 - 18, find of the exact solutions of the equation and then list those solutions which are in the interval \([0, 2\pi)\).

1. \(\sin \left( 5x \right) = 0\)
2. \(\cos \left( 3x \right) = \dfrac{1}{2}\)
3. \(\sin \left( -2x \right) = \dfrac{\sqrt{3}}{2}\)
4. \(\tan \left( 6x \right) = 1\)
5. \(\csc \left( 4x \right) = -1\)
6. \(\sec \left( 3x \right) = \sqrt{2}\)
7. \(\cot \left( 2x \right) = -\dfrac{\sqrt{3}}{3}\)
8. \(\cos \left( 9x \right) = 9\)
9. \(\sin \left( \dfrac{x}{3} \right) = \dfrac{\sqrt{2}}{2}\)
10. \(\cos \left( x + \dfrac{5\pi}{6} \right) = 0\)
11. \(\sin \left( 2x - \dfrac{\pi}{3} \right) = -\dfrac{1}{2}\)
12. \(2\cos \left( x + \dfrac{7\pi}{4} \right) = \sqrt{3}\)
13. \(\csc(x) = 0\)
14. \(\tan \left( 2x - \pi \right) = 1\)
15. \(\tan^{2} \left( x \right) = 3\)
16. \(\sec^{2} \left( x \right) = \dfrac{4}{3}\)
17. \(\cos^{2} \left( x \right) = \dfrac{1}{2}\)
18. \(\sin^{2} \left( x \right) = \dfrac{3}{4}\)

In Exercises 19 - 42, solve the equation, giving the exact solutions which lie in \([0, 2\pi)\)

19. \(\sin \left( x \right) = \cos \left( x \right)\)
20. \(\sin \left( 2x \right) = \sin \left( x \right)\)
21. \(\sin \left( 2x \right) = \cos \left( x \right)\)
22. \(\cos \left( 2x \right) = \sin \left( x \right)\)
23. \(\cos \left( 2x \right) = \cos \left( x \right)\)
24. \(\cos(2x) = 2 - 5\cos(x)\)
25. \(3\cos(2x) + \cos(x) + 2 = 0\)
26. \(\cos(2x) = 5\sin(x) - 2\)
27. \(3\cos(2x) = \sin(x) + 2\)
28. \(2\sec^{2}(x) = 3 - \tan(x)\)
29. \(\tan^{2}(x) = 1-\sec(x)\)
30. \(\cot^{2}(x) = 3\csc(x) - 3\)
31. \(\sec(x) = 2\csc(x)\)
32. \(\cos(x)\csc(x)\cot(x) = 6-\cot^{2}(x)\)
33. \(\sin(2x) = \tan(x)\)
34. \(\cot^{4}(x) = 4\csc^{2}(x) - 7\)
35. \(\cos(2x) + \csc^{2}(x) = 0\)
36. \(\tan^{3} \left( x \right) = 3\tan \left( x \right)\)
37. \(\tan^{2} \left( x \right) = \dfrac{3}{2} \sec \left( x \right)\)
38. \(\cos^{3} \left( x \right) = -\cos \left( x \right)\)
39. \(\tan (2x) - 2\cos(x) = 0\)
40. \(\csc^{3}(x) + \csc^{2}(x) = 4\csc(x) + 4\)
41. \(2\tan(x) = 1 - \tan^{2}(x)\)
42. \(\tan \left( x \right) = \sec \left( x \right)\)

In Exercises 43 - 58, solve the equation, giving the exact solutions which lie in \([0, 2\pi)\)

43. \(\sin(6x) \cos(x) = -\cos(6x) \sin(x)\)
44. \(\sin(3x)\cos(x) = \cos(3x) \sin(x)\)
45. \(\cos(2x)\cos(x) + \sin(2x)\sin(x) = 1\)
46. \(\cos(5x)\cos(3x) - \sin(5x)\sin(3x) = \dfrac{\sqrt{3}}{2}\)
47. \(\sin(x) + \cos(x) = 1\)
48. \(\sin(x) + \sqrt{3} \cos(x) = 1\)
49. \(\sqrt{2} \cos(x) - \sqrt{2} \sin(x) = 1\)
50. \(\sqrt{3} \sin(2x) + \cos(2x) = 1\)
51. \(\cos(2x) - \sqrt{3} \sin(2x) = \sqrt{2}\)
52. \(3\sqrt{3}\sin(3x) - 3\cos(3x) = 3\sqrt{3}\)
53. \(\cos(3x) = \cos(5x)\)
54. \(\cos(4x) = \cos(2x)\)
55. \(\sin(5x) = \sin(3x)\)
56. \(\cos(5x) = -\cos(2x)\)
57. \(\sin(6x) + \sin(x) = 0\)
58. \(\tan(x) = \cos(x)\)

In Exercises 59 - 68, solve the equation.

59. \(\arccos (2 x)=\pi\)
60. \(\pi-2 \arcsin (x)=2 \pi\)
61. \(4 \arctan (3 x-1)-\pi=0\)
62. \(6 \operatorname{arccot}(2 x)-5 \pi=0\)
63. \(4 \operatorname{arcsec}\left(\frac{x}{2}\right)=\pi\)
64. \(12 \operatorname{arccsc}\left(\frac{x}{3}\right)=2 \pi\)
65. \(9 \arcsin ^{2}(x)-\pi^{2}=0\)
66. \(9 \arccos ^{2}(x)-\pi^{2}=0\)
67. \(8 \operatorname{arccot}^{2}(x)+3 \pi^{2}=10 \pi \operatorname{arccot}(x)\)
68. \(6 \arctan (x)^{2}=\pi \arctan (x)+\pi^{2}\)

In Exercises 69 - 80, solve the inequality. Express the exact answer in interval notation, restricting your attention \(0 \leq x \leq 2 \pi\).

69. \(\sin \left( x \right) \leq 0\)
70. \(\tan \left( x \right) \geq \sqrt{3}\)
71. \(\sec^{2} \left( x \right) \leq 4\)
72. \(\cos^{2} \left( x \right) > \dfrac{1}{2}\)
73. \(\cos \left( 2x \right) \leq 0\)
74. \(\sin \left( x + \dfrac{\pi}{3} \right) > \dfrac{1}{2}\)
75. \(\cot^{2} \left( x \right) \geq \dfrac{1}{3}\)
76. \(2\cos(x) \geq 1\)
77. \(\sin(5x) \geq 5\)
78. \(\cos(3x) \leq 1\)
79. \(\sec(x) \leq \sqrt{2}\)
80. \(\cot(x) \leq 4\)

In Exercises 81 - 86, solve the inequality. Express the exact answer in interval notation, restricting your attention to \(-\pi \leq x \leq \pi\).

81. \(\cos \left( x \right) > \dfrac{\sqrt{3}}{2}\)
82. \(\sin(x) > \dfrac{1}{3}\)
83. \(\sec \left( x \right) \leq 2\)
84. \(\sin^{2} \left( x \right) < \dfrac{3}{4}\)
85. \(\cot \left( x \right) \geq -1\)
86. \(\cos(x) \geq \sin(x)\)

In Exercises 87 - 92, solve the inequality. Express the exact answer in interval notation, restricting your attention to \(-2 \pi \leq x \leq 2 \pi\).

87. \(\csc \left( x \right) > 1\)
88. \(\cos(x) \leq \dfrac{5}{3}\)
89. \(\cot(x) \geq 5\)
90. \(\tan^{2} \left( x \right) \geq 1\)
91. \(\sin(2x) \geq \sin(x)\)
92. \(\cos(2x) \leq \sin(x)\)

In Exercises 93 - 92, solve the given inequality.

93. \(\arcsin (2 x)>0\)
94. \(3 \arccos (x) \leq \pi\)
95. \(6 \operatorname{arccot}(7 x) \geq \pi\)
96. \(\pi>2 \arctan (x)\)
97. \(2 \arcsin (x)^{2}>\pi \arcsin (x)\)
98. \(12 \arccos (x)^{2}+2 \pi^{2}>11 \pi \arccos (x)\)

In Exercises 99 - 107, express the domain of the function using the extended interval notation.

99. \(f(x)=\frac{1}{\cos (x)-1}\)
100. \(f(x)=\frac{\cos (x)}{\sin (x)+1}\)
101. \(f(x)=\sqrt{\tan ^{2}(x)-1}\)
102. \(f(x)=\sqrt{2-\sec (x)}\)
103. \(f(x)=\csc (2 x)\)
104. \(f(x)=\frac{\sin (x)}{2+\cos (x)}\)
105. \(f(x)=3 \csc (x)+4 \sec (x)\)
106. \(f(x)=\ln (|\cos (x)|)\)
107. \(f(x)=\arcsin (\tan (x))\)
108. With the help of your classmates, determine the number of solutions to \(\sin(x) = \frac{1}{2}\) in \([0,2\pi)\). Then find the number of solutions to \(\sin(2x) = \frac{1}{2}\), \(\sin(3x) = \frac{1}{2}\) and \(\sin(4x) = \frac{1}{2}\) in \([0,2\pi)\). A pattern should emerge. Explain how this pattern would help you solve equations like \(\sin(11x) = \frac{1}{2}\). Now consider \(\sin\left(\frac{x}{2}\right) = \frac{1}{2}\), \(\sin\left(\frac{3x}{2}\right) = \frac{1}{2}\) and \(\sin\left(\frac{5x}{2}\right) = \frac{1}{2}\). What do you find? Replace \(\dfrac{1}{2}\) with \(-1\) and repeat the whole exploration.

Answers

1. \(x = \dfrac{\pi k}{5}; \; x = 0, \dfrac{\pi}{5}, \dfrac{2\pi}{5}, \dfrac{3\pi}{5}, \dfrac{4\pi}{5}, \pi, \dfrac{6\pi}{5}, \dfrac{7\pi}{5}, \dfrac{8\pi}{5}, \dfrac{9\pi}{5}\)
2. \(x = \dfrac{\pi}{9} + \dfrac{2\pi k}{3}\) or \(x = \dfrac{5\pi}{9} + \dfrac{2\pi k}{3}; \; x = \dfrac{\pi}{9}, \dfrac{5\pi}{9}, \dfrac{7\pi}{9}, \dfrac{11\pi}{9}, \dfrac{13\pi}{9}, \dfrac{17\pi}{9}\)
3. \(x = \dfrac{2\pi}{3} + \pi k\) or \(x = \dfrac{5\pi}{6} + \pi k; \; x = \dfrac{2\pi}{3}, \dfrac{5\pi}{6}, \dfrac{5\pi}{3}, \dfrac{11\pi}{6}\)
4. \(x = \dfrac{\pi}{24} + \dfrac{\pi k}{6}; \; x = \dfrac{\pi}{24}, \dfrac{5\pi}{24}, \dfrac{3\pi}{8}, \dfrac{13\pi}{24}, \dfrac{17\pi}{24}, \dfrac{7\pi}{8}, \dfrac{25\pi}{24}, \dfrac{29\pi}{24}, \dfrac{11\pi}{8}, \dfrac{37\pi}{24}, \dfrac{41\pi}{24}, \dfrac{15\pi}{8}\)
5. \(x = \dfrac{3\pi}{8} + \dfrac{\pi k}{2}; \; x = \dfrac{3\pi}{8}, \dfrac{7\pi}{8}, \dfrac{11\pi}{8}, \dfrac{15\pi}{8}\)
6. \(x = \dfrac{\pi}{12} + \dfrac{2\pi k}{3}\) or \(x = \dfrac{7\pi}{12} + \dfrac{2\pi k}{3}; \; x = \dfrac{\pi}{12}, \dfrac{7\pi}{12}, \dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{17\pi}{12}, \dfrac{23\pi}{12}\)
7. \(x = \dfrac{\pi}{3} + \dfrac{\pi k}{2}; \; x = \dfrac{\pi}{3}, \dfrac{5\pi}{6}, \dfrac{4\pi}{3}, \dfrac{11\pi}{6}\)
8. No solution
9. \(x = \dfrac{3\pi}{4} + 6\pi k\) or \(x = \dfrac{9\pi}{4} + 6\pi k; \; x = \dfrac{3\pi}{4}\)
10. \(x = -\dfrac{\pi}{3} + \pi k; \; x = \dfrac{2\pi}{3}, \dfrac{5\pi}{3}\)
11. \(x = \dfrac{3\pi}{4} + \pi k\) or \(x = \dfrac{13\pi}{12} + \pi k; \; x = \dfrac{\pi}{12}, \dfrac{3\pi}{4}, \dfrac{13\pi}{12}, \dfrac{7\pi}{4}\)
12. \(x = -\dfrac{19\pi}{12} + 2\pi k\) or \(x = \dfrac{\pi}{12} + 2\pi k; \; x = \dfrac{\pi}{12}, \dfrac{5\pi}{12}\)
13. No solution
14. \(x = \dfrac{5\pi}{8} + \dfrac{\pi k}{2}; \; x = \dfrac{\pi}{8}, \dfrac{5\pi}{8}, \dfrac{9\pi}{8}, \dfrac{13\pi}{8}\)
15. \(x = \dfrac{\pi}{3} + \pi k\) or \(x = \dfrac{2\pi}{3} + \pi k; \; x = \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}\)
16. \(x = \dfrac{\pi}{6} + \pi k\) or \(x = \dfrac{5\pi}{6} + \pi k; \; x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{11\pi}{6}\)
17. \(x = \dfrac{\pi}{4} + \dfrac{\pi k}{2}; \; x = \dfrac{\pi}{4}, \dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4}\)
18. \(x = \dfrac{\pi}{3} + \pi k\) or \(x = \dfrac{2\pi}{3} + \pi k; \; x = \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}\)
19. \(x = \dfrac{\pi}{4}, \dfrac{5\pi}{4}\)
20. \(x = 0, \dfrac{\pi}{3}, \pi, \dfrac{5\pi}{3}\)
21. \(x = \dfrac{\pi}{6}, \dfrac{\pi}{2}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}\)
22. \(x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}\)
23. \(x = 0, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}\)
24. \(x=\dfrac{\pi}{3}, \dfrac{5\pi}{3}\)
25. \(x = \dfrac{2\pi}{3}, \dfrac{4\pi}{3}, \arccos\left(\dfrac{1}{3}\right), 2\pi -\arccos\left(\dfrac{1}{3}\right)\)
26. \(x=\dfrac{\pi}{6}, \dfrac{5\pi}{6}\)
27. \(x = \dfrac{7\pi}{6}, \dfrac{11\pi}{6}, \arcsin\left(\dfrac{1}{3}\right), \pi - \arcsin\left(\dfrac{1}{3}\right)\)
28. \(x=\dfrac{3\pi}{4}, \dfrac{7\pi}{4}, \arctan\left(\dfrac{1}{2}\right), \pi +\arctan\left(\dfrac{1}{2}\right)\)
29. \(x=0, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}\)
30. \(x=\dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{\pi}{2}\)
31. \(x=\arctan(2), \pi + \arctan(2)\)
32. \(x = \dfrac{\pi}{6}, \dfrac{7\pi}{6}, \dfrac{5\pi}{6}, \dfrac{11\pi}{6}\)
33. \(x = 0, \pi, \dfrac{\pi}{4}, \dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4}\)
34. \(x = \dfrac{\pi}{6}, \dfrac{\pi}{4}, \dfrac{3\pi}{4}, \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4}, \dfrac{11\pi}{6}\)
35. \(x = \dfrac{\pi}{2}, \dfrac{3\pi}{2}\)
36. \(x = 0, \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \pi, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}\)
37. \(x = \dfrac{\pi}{3}, \dfrac{5\pi}{3}\)
38. \(x = \dfrac{\pi}{2}, \dfrac{3\pi}{2}\)
39. \(x = \dfrac{\pi}{6}, \dfrac{\pi}{2}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}\)
40. \(x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{3\pi}{2}, \dfrac{11\pi}{6}\)
41. \(x = \dfrac{\pi}{8}, \dfrac{5\pi}{8}, \dfrac{9\pi}{8}, \dfrac{13\pi}{8}\)
42. No solution
43. \(x = 0, \dfrac{\pi}{7}, \dfrac{2\pi}{7}, \dfrac{3\pi}{7}, \dfrac{4\pi}{7}, \dfrac{5\pi}{7}, \dfrac{6\pi}{7}, \pi, \dfrac{8\pi}{7}, \dfrac{9\pi}{7}, \dfrac{10\pi}{7}, \dfrac{11\pi}{7}, \dfrac{12\pi}{7}, \dfrac{13\pi}{7}\)
44. \(x=0, \dfrac{\pi}{2}, \pi, \dfrac{3\pi}{2}\)
45. \(x = 0\)
46. \(x = \dfrac{\pi}{48}, \dfrac{11\pi}{48}, \dfrac{13\pi}{48}, \dfrac{23\pi}{48}, \dfrac{25\pi}{48}, \dfrac{35\pi}{48}, \dfrac{37\pi}{48}, \dfrac{47\pi}{48}, \dfrac{49\pi}{48}, \dfrac{59\pi}{48}, \dfrac{61\pi}{48}, \dfrac{71\pi}{48}, \dfrac{73\pi}{48}, \dfrac{83\pi}{48}, \dfrac{85\pi}{48}, \dfrac{95\pi}{48}\)
47. \(x = 0, \dfrac{\pi}{2}\)
48. \(x = \dfrac{\pi}{2}, \dfrac{11\pi}{6}\)
49. \(x = \dfrac{\pi}{12}, \dfrac{17\pi}{12}\)
50. \(x= 0, \pi, \dfrac{\pi}{3}, \dfrac{4\pi}{3}\)
51. \(x = \dfrac{17 \pi}{24}, \dfrac{41 \pi}{24}, \dfrac{23\pi}{24}, \dfrac{47\pi}{24}\)
52. \(x = \dfrac{\pi}{6}, \dfrac{5\pi}{18}, \dfrac{5\pi}{6}, \dfrac{17\pi}{18}, \dfrac{3\pi}{2}, \dfrac{29\pi}{18}\)
53. \(x = 0, \dfrac{\pi}{4}, \dfrac{\pi}{2}, \dfrac{3\pi}{4}, \pi, \dfrac{5\pi}{4}, \dfrac{3\pi}{2}, \dfrac{7\pi}{4}\)
54. \(x = 0, \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \pi, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}\)
55. \(x = 0, \dfrac{\pi}{8}, \dfrac{3\pi}{8}, \dfrac{5\pi}{8}, \dfrac{7\pi}{8}, \pi, \dfrac{9\pi}{8}, \dfrac{11\pi}{8}, \dfrac{13\pi}{8}, \dfrac{15\pi}{8}\)
56. \(x = \dfrac{\pi}{7}, \dfrac{\pi}{3}, \dfrac{3\pi}{7}, \dfrac{5\pi}{7}, \pi, \dfrac{9\pi}{7}, \dfrac{11\pi}{7}, \dfrac{5\pi}{3}, \dfrac{13\pi}{7}\)
57. \(x = \dfrac{2\pi}{7}, \dfrac{4\pi}{7}, \dfrac{6\pi}{7}, \dfrac{8\pi}{7}, \dfrac{10\pi}{7}, \dfrac{12\pi}{7}, \dfrac{\pi}{5}, \dfrac{3\pi}{5}, \pi, \dfrac{7\pi}{5}, \dfrac{9\pi}{5}\)
58. \(x = \arcsin \left( \dfrac{-1 + \sqrt{5}}{2} \right) \approx 0.6662, \pi - \arcsin \left( \dfrac{-1 + \sqrt{5}}{2} \right) \approx 2.4754\)
59. \(x=-\frac{1}{2}\)
60. \(x = −1\)
61. \(x=\frac{2}{3}\)
62. \(x=-\frac{\sqrt{3}}{2}\)
63. \(x=2 \sqrt{2}\)
64. \(x = 6\)
65. \(x=\pm \frac{\sqrt{3}}{2}\)
66. \(x=\frac{1}{2}\)
67. \(x = −1, 0\)
68. \(x=-\sqrt{3}\)
69. \(\left[ \pi, 2\pi \right]\)
70. \(\left[ \dfrac{\pi}{3}, \dfrac{\pi}{2} \right) \cup \left[ \dfrac{4\pi}{3}, \dfrac{3\pi}{2} \right)\)
71. \(\left[ 0, \dfrac{\pi}{3} \right] \cup \left[ \dfrac{2\pi}{3}, \dfrac{4\pi}{3} \right] \cup \left[ \dfrac{5\pi}{3}, 2\pi \right]\)
72. \(\left[ 0, \dfrac{\pi}{4} \right) \cup \left( \dfrac{3\pi}{4}, \dfrac{5\pi}{4} \right) \cup \left( \dfrac{7\pi}{4}, 2\pi \right]\)
73. \(\left[ \dfrac{\pi}{4}, \dfrac{3\pi}{4} \right] \cup \left[ \dfrac{5\pi}{4}, \dfrac{7\pi}{4} \right]\)
74. \(\left[ 0, \dfrac{\pi}{2} \right) \cup \left( \dfrac{11\pi}{6}, 2\pi \right]\)
75. \(\left( 0, \dfrac{\pi}{3} \right] \cup \left[ \dfrac{2\pi}{3}, \pi \right) \cup \left( \pi, \dfrac{4\pi}{3} \right] \cup \left[ \dfrac{5\pi}{3}, 2\pi \right)\)
76. \(\left[0, \dfrac{\pi}{3}\right] \cup \left[\dfrac{5\pi}{3}, 2\pi\right]\)
77. No solution
78. \([0, 2\pi]\)
79. \(\left[0, \dfrac{\pi}{4} \right] \cup \left(\dfrac{\pi}{2}, \dfrac{3\pi}{2}\right) \cup \left[\dfrac{7\pi}{4}, 2\pi\right]\)
80. \(\left[\text{arccot}(4), \pi \right) \cup \left[ \pi + \text{arccot}(4), 2\pi\right)\)
81. \(\left( -\dfrac{\pi}{6}, \dfrac{\pi}{6} \right)\)
82. \(\left( \arcsin\left(\dfrac{1}{3}\right), \pi - \arcsin\left(\dfrac{1}{3}\right) \right)\)
83. \(\left[ -\pi, -\dfrac{\pi}{2} \right) \cup \left[ -\dfrac{\pi}{3}, \dfrac{\pi}{3} \right] \cup \left( \dfrac{\pi}{2}, \pi \right]\)
84. \(\left( -\dfrac{2\pi}{3}, -\dfrac{\pi}{3} \right) \cup \left( \dfrac{\pi}{3}, \dfrac{2\pi}{3} \right)\)
85. \(\left( -\pi, -\dfrac{\pi}{4} \right] \cup \left( 0, \dfrac{3\pi}{4} \right]\)
86. \(\left[ -\dfrac{3\pi}{4}, \dfrac{\pi}{4} \right]\)
87. \(\left( -2\pi, -\dfrac{3\pi}{2} \right) \cup \left( -\dfrac{3\pi}{2}, -\pi \right) \cup \left( 0, \dfrac{\pi}{2} \right) \cup \left( \dfrac{\pi}{2}, \pi \right)\)
88. \([-2\pi, 2\pi]\)
89. \(\left(-2\pi, \text{arccot}(5) - 2\pi\right] \cup \left(-\pi, \text{arccot}(5) - \pi\right] \cup \left(0, \text{arccot}(5)\right] \cup \left(\pi, \pi + \text{arccot}(5)\right]\)
90. \(\left[ -\dfrac{7\pi}{4}, -\dfrac{3\pi}{2} \right) \cup \left( -\dfrac{3\pi}{2}, -\dfrac{5\pi}{4} \right] \cup \left[ -\dfrac{3\pi}{4}, -\dfrac{\pi}{2} \right) \cup \left( -\dfrac{\pi}{2}, -\dfrac{\pi}{4} \right] \cup \left[ \dfrac{\pi}{4}, \dfrac{\pi}{2} \right) \cup \left( \dfrac{\pi}{2}, \dfrac{3\pi}{4} \right] \cup \left[ \dfrac{5\pi}{4}, \dfrac{3\pi}{2} \right) \cup \left( \dfrac{3\pi}{2}, \dfrac{7\pi}{4} \right]\)
91. \(\left[ -2\pi, -\dfrac{5\pi}{3} \right] \cup \left[ -\pi, -\dfrac{\pi}{3} \right] \cup \left[ 0, \dfrac{\pi}{3} \right] \cup \left[ \pi, \dfrac{5\pi}{3} \right]\)
92. \(\left[-\frac{11 \pi}{6},-\frac{7 \pi}{6}\right] \cup\left[\frac{\pi}{6}, \frac{5 \pi}{6}\right] \cup,\left\{-\frac{\pi}{2}, \frac{3 \pi}{2}\right\}\)
93. \(\left(0, \frac{1}{2}\right]\)
94. \(\left[\frac{1}{2}, 1\right]\)
95. \(\left(-\infty, \frac{\sqrt{3}}{7}\right]\)
96. \((-\infty, \infty)\)
97. \([-1,0)\)
98. \(\left[-1,-\frac{1}{2}\right) \cup\left(\frac{\sqrt{2}}{2}, 1\right]\)
99. \(\bigcup_{k=-\infty}^{\infty}(2 k \pi,(2 k+2) \pi)\)
100. \(\bigcup_{k=-\infty}^{\infty}\left(\frac{(4 k-1) \pi}{2}, \frac{(4 k+3) \pi}{2}\right)\)
101. \(\bigcup_{k=-\infty}^{\infty}\left\{\left[\frac{(4 k+1) \pi}{4}, \frac{(2 k+1) \pi}{2}\right) \cup\left(\frac{(2 k+1) \pi}{2}, \frac{(4 k+3) \pi}{4}\right]\right\}\)
102. \(\bigcup_{k=-\infty}^{\infty}\left\{\left[\frac{(6 k-1) \pi}{3}, \frac{(6 k+1) \pi}{3}\right] \cup\left(\frac{(4 k+1) \pi}{2}, \frac{(4 k+3) \pi}{2}\right)\right\}\)
103. \(\bigcup_{k=-\infty}^{\infty}\left(\frac{k \pi}{2}, \frac{(k+1) \pi}{2}\right)\)
104. \((-\infty, \infty)\)
105. \(\bigcup_{k=-\infty}^{\infty}\left(\frac{k \pi}{2}, \frac{(k+1) \pi}{2}\right)\)
106. \(\bigcup_{k=-\infty}^{\infty}\left(\frac{(2 k-1) \pi}{2}, \frac{(2 k+1) \pi}{2}\right)\)
107. \(\bigcup_{k=-\infty}^{\infty}\left[\frac{(4 k-1) \pi}{4}, \frac{(4 k+1) \pi}{4}\right]\)