6.2e: Exercises - Trig Equations

Give all answers in radians unless otherwise indicated.   

A: Concepts

Exercise \(\PageIndex{A}\)

1. Will there always be solutions to trigonometric function equations? If not, describe an equation that would not have a solution. Explain why or why not.

2. When solving a trigonometric equation involving more than one trig function, do we always want to try to rewrite the equation so it is expressed in terms of one trigonometric function? Why or why not?

3. When solving linear trig equations in terms of only sine or cosine, how do we know whether there will be solutions?

Answers to odd exercises.

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

3. Rewrite the equation in the form \( \sin(u) = c\) or \( \cos(u) = c\). If \( |c| \le 1,\) then the equation will have a solution; otherwise it will not have a solution. 

B: Linear Equations - Special Angle Solutions 

Exercise \(\PageIndex{B}\)

\( \bigstar \) Find all solutions on the interval \(0\le \theta <2\pi\).

5. \(2\sin \left(\theta \right)=-\sqrt{2} \\[2pt] \) 6. \(2\sin \left(\theta \right)=\sqrt{3} \\[2pt] \) 7. \(\sin \left(\theta \right)=1 \\[2pt] \) 8. \(\sin \left(\theta \right)=0\)

9. \(2\cos \left(\theta \right)=1 \\[2pt] \) 10. \(2\cos \left(\theta \right)=-\sqrt{2} \\[2pt] \) 11. \(\cos \left(\theta \right)=0 \\[2pt] \) 12. \(\cos \left(\theta \right)=1\)

13. \(\tan x=1 \\[2pt] \) 14. \(\tan \theta=-1 \\[2pt] \) 15. \( \sqrt{3}\tan \theta=1 \\[2pt] \) 16. \( \sqrt{3} +\tan \theta=0\)

17. \( \sqrt{3} \csc x+2=  0 \\[2pt]\) 18. \(  \sqrt{3}\sec \theta+2 = 0 \\[2pt] \) 19. \(  \cot x+1=0\)

\( \bigstar \) State the general solution and the first four non-negative solutions.

21. \(2\sin \left(\theta \right) = -1 \\[2pt]  \) 22. \(2\sin \left(\theta \right) = 1 \\[2pt] \) 23. \(2\sin \left(3\theta \right)= \sqrt{2} \\[2pt] \) 24. \(2\sin \left(3\pi\theta \right)=-\sqrt{2} \\[2pt] \) 25. \(2\sin \left(\dfrac{\theta}{2} \right)=\sqrt{3} \\[2pt] \) 26. \(\sin \left(\dfrac{2\pi }{3} \theta \right)=-1 \\[2pt] \) 27. \(\sin \left(\dfrac{3\pi }{4} \theta  + \dfrac{\pi}{2}\right)=0 \\[2pt] \) 28. \( 2 \sin \left( 4 \pi \theta -\dfrac{\pi}{3} \right)= -\sqrt{3} \\[2pt] \) 29. \( \csc \left(2\pi x \right)=-2 \\[2pt] \) 30. \( \csc \left(\dfrac{x }{3} \right)=\sqrt{ 2} \\[2pt] \)

31. \(2\cos \left(\theta \right)=\sqrt{2} \\[2pt] \) 32. \(2\cos \left(\theta \right)=-1 \\[2pt] \) 33. \(2\cos \left(2 \pi \theta \right)=1 \\[2pt] \) 34. \(2\cos \left(3 \theta \right)=-\sqrt{2} \\[2pt] \) 35. \(\cos \left(\dfrac{\pi }{4} \theta \right)=-1 \\[2pt] \) 36. \(2\cos \left( \dfrac{\theta}{3} \right)=\sqrt{3} \\[2pt] \) 37. \(  2 \cos \left(3 \pi \theta - \dfrac{\pi}{4} \right)=-\sqrt{3}  \\[2pt]  \) 38. \(\cos \left(\dfrac{\theta}{2} +\dfrac{\pi}{6}  \right)=0 \\[2pt] \) 39. \(\sec \left(  3x \right)=-2 \\[2pt] \) 40. \( \sec \left(\dfrac{\pi }{5} x \right)=\sqrt{2} \\[2pt] \)

41. \( \sqrt{3} \tan ( 3 x ) = -1 \\[2pt] \) 42. \(  \tan (  \pi x ) = \sqrt{3} \\[2pt] \) 43. \( \tan\left(\dfrac{\theta}{3} \right) = 1 \\[2pt] \) 44. \( \tan\left(\dfrac{2\pi }{3} \theta \right) = -1 \\[2pt] \) 45. \(  \tan\left(2x+ \dfrac{\pi }{2}  \right) = -\sqrt{3}  \\[2pt] \) 46. \(  \tan\left(\dfrac{\pi }{6} \theta -\dfrac{\pi}{9}\right) = 0  \\[2pt] \) 47. \( \cot (2 \pi x ) = -1 \\[2pt]  \) 48. \(  \cot \left(\dfrac{\pi }{6} x \right) = \sqrt{3}  \\[2pt] \)

Answers to odd exercises.

5. \(\dfrac{5\pi}{4}\), \(\dfrac{7\pi}{4}\)     7. \(\dfrac{\pi}{2}\)     9. \(\dfrac{\pi}{3}\), \(\dfrac{5\pi}{3}\)     11. \(\dfrac{\pi}{2}\), \(\dfrac{3\pi}{2}\)     13. \(\dfrac{\pi }{4}, \dfrac{5\pi }{4}\)     15.  \(\dfrac{\pi }{6}, \dfrac{7\pi }{6}\)    17. \(\dfrac{4\pi}{3}\), \(\dfrac{5\pi}{3}\)      19. \(\dfrac{3\pi }{4}, \dfrac{7\pi }{4}\)

\(k\)  below represents any integer         

21. \(\dfrac{7 \pi}{6} + 2 \pi k\), \(\dfrac{11\pi}{6} + 2 \pi k\);  \( \quad\)  \( \dfrac{7 \pi}{6},\)  \(\dfrac{11\pi}{6},\)   \( \dfrac{19 \pi}{6},\)  \(\dfrac{23\pi}{6} \\[2pt] \)      23. \(\dfrac{ \pi}{12} + \dfrac{ 2\pi}{3} k\), \(\dfrac{ \pi}{4} + \dfrac{ 2\pi}{3} k\);  \( \quad\)  \( \dfrac{ \pi}{12},\)  \(\dfrac{\pi}{4},\)   \( \dfrac{3 \pi}{4},\)  \(\dfrac{11\pi}{12} \\[2pt] \)  25. \(\dfrac{2 \pi}{3} + 4 \pi k\), \(\dfrac{4\pi}{3} + 4 \pi k\);  \( \quad\)  \( \dfrac{2 \pi}{3},\)  \(\dfrac{4\pi}{3},\)   \( \dfrac{14 \pi}{3},\)  \(\dfrac{16\pi}{3} \\[2pt] \)  27. \(-\dfrac{2 }{3} + \dfrac{4}{3}k \);  \( \quad\)  \( \dfrac{2}{3},\)  \(2,\)    \( \dfrac{10}{3},\)  \( \dfrac{14}{3} \\[2pt] \)  29. \(\dfrac{7}{12} + k \),  \( \dfrac{11}{12} + k \);  \( \quad\)  \( \dfrac{7}{12}, \)   \( \dfrac{11}{12}, \)   \( \dfrac{19}{12}, \)   \( \dfrac{23}{12}  \\[2pt] \) 31. \(\dfrac{\pi}{4} + 2 \pi k\), \(\dfrac{7\pi}{4} + 2 \pi k\);  \( \quad\)  \( \dfrac{\pi}{4}, \)   \( \dfrac{7\pi}{4}, \)   \( \dfrac{9\pi}{4}, \)   \( \dfrac{15\pi}{4} \\[2pt]  \) 33. \(\dfrac{1}{6} +  k\), \(\dfrac{5}{6} + k\);  \( \quad\)  \( \dfrac{1}{6}, \)  \( \dfrac{5}{6}, \)  \( \dfrac{7}{6}, \)  \( \dfrac{11}{6}  \\[2pt] \)

35.  \( 4 +  8k\);     \( 4\), \( 12\),  \( 20\),  \( 28 \\[2pt] \) 37.  \(\dfrac{13}{36} + \dfrac{2 }{3} k\), \(\dfrac{17}{36} + \dfrac{2 }{3} k\);  \( \quad\)  \( \dfrac{13}{36},\)  \(\dfrac{17}{36},\)    \( \dfrac{37}{36},\)  \( \dfrac{41}{36} \\[2pt] \) 39. \(\dfrac{2\pi}{9} + \dfrac{2 \pi}{3} k\), \(\dfrac{4\pi}{9} + \dfrac{2 \pi}{3} k\);  \( \quad\)  \( \dfrac{2 \pi}{9},\)   \( \dfrac{4 \pi}{9},\)   \( \dfrac{8 \pi}{9},\)   \( \dfrac{10 \pi}{9} \\[2pt] \)  41. \(- \dfrac{\pi}{18} + \dfrac{\pi}{3} k\);  \( \quad\)  \( \dfrac{5 \pi}{18},\)   \( \dfrac{ 11\pi}{18},\)   \( \dfrac{17 \pi}{18},\)   \( \dfrac{ 23\pi}{18} \\[2pt] \) 43. \(\dfrac{3\pi}{4} + 3 \pi k\);  \( \quad\)  \( \dfrac{3\pi}{4}, \)   \( \dfrac{15\pi}{4}, \)   \( \dfrac{27\pi}{4}, \)   \( \dfrac{39\pi}{4} \\[2pt]  \)   45.  \(\dfrac{\pi}{12} + \dfrac{\pi}{2} k\);  \( \quad\)  \( \dfrac{\pi}{12}, \)   \( \dfrac{7\pi}{12}, \)   \( \dfrac{13\pi}{12}, \)   \( 19\dfrac{\pi}{12} \\[2pt]  \)     47.  \(\dfrac{3}{8} + \dfrac{k}{2} \);  \( \quad\)  \( \dfrac{3}{8}, \) \( \dfrac{7}{8}, \) \( \dfrac{11}{8}, \) \( \dfrac{15}{8} \\[2pt] \)

C: Quadratic Equations - Special Angle Solutions

Exercise \(\PageIndex{C}\)

\( \bigstar \) Find all solutions on the interval \([0, 2\pi )\). Give exact answers.

51. \(\sin ^{2} x=\dfrac{1}{4}\) 52. \(4\sin^2 x-2=0\) 53. \(\sin^3 t=\sin t\) 54. \(\sin^2 x+\sin x-2=0\) 55. \(2\sin ^{2} w+3\sin w+1=0\)  56. \(2\sin ^{2} x+3\sin x-2=0\)  57. \(\csc^2 x-4=0\)

59. \(\cos ^{2} \theta =\dfrac{1}{2}\) 60. \(4\cos^2 x-3=0\) 61. \(\cos^3 t=\cos t\) 62. \(2\cos ^{2} t+\cos (t)=1\)  63. \(\cos^2 x-2\cos x-3=0\) 64. \(6\cos ^{2}(\theta )=4-5\cos (\theta )\) 65. \(\sec^2 x =1\)

67 \( \tan^2 \theta = 3 \) 68. \( 3\tan^2 \theta = 1 \) 69. \(\tan ^{3} (x)=3\tan (x)\) 70. \(\tan ^{5} (x)=\tan (x)\) 71. \( 9 \tan ^{5}(x)-\tan (x)=0\) 72. \(\tan^2 x-\sqrt{3}\tan x=0\) 73. \(\cot^2 x=-\cot x\)

Answers to odd exercises

51. \(\dfrac{\pi}{6}\),  \(\dfrac{5\pi}{6}\),  \(\dfrac{7\pi}{6}\),  \(\dfrac{11\pi}{6}\) 53.  \(0, \; \pi, \; \dfrac{\pi}{2}, \; \dfrac{3\pi}{2} \) 55. \(\dfrac{3\pi}{2}\),  \(\dfrac{7\pi}{6}\),  \(\dfrac{11\pi}{6}\)	57. \(\dfrac{\pi }{6}, \; \dfrac{5\pi }{6}, \; \dfrac{7\pi }{6}, \; \dfrac{11\pi }{6}\) 59. \(\dfrac{\pi }{4}, \; \dfrac{3\pi }{4}, \; \dfrac{5\pi }{4}, \; \dfrac{7\pi }{4}\) 61. \(0, \; \dfrac{\pi }{2}, \; \pi , \; \dfrac{3\pi }{2}\)	63. \(\pi\) 65. \(0, \;  \pi \)  67. \(\dfrac{\pi}{3}\),  \(\dfrac{2\pi}{3},  \dfrac{4\pi}{3}\), \(\dfrac{5\pi}{3}\) 69. 0, \(\dfrac{\pi}{3}\), \(\dfrac{2\pi}{3}\), \(\pi\), \(\dfrac{4\pi}{3}\), \(\dfrac{5\pi}{3}\)	71.  0, \(\dfrac{\pi}{6}\), \(\dfrac{5\pi}{6}\), \(\pi\), \(\dfrac{7\pi}{6}\), \(\dfrac{11\pi}{6}\) 73. \( \dfrac{\pi }{2} \),  \(\dfrac{3\pi }{2} \),  \( \dfrac{3\pi }{4}, \) \( \dfrac{7\pi }{4}\)

\( \bigstar \) Find all solutions on the interval \([0, 2\pi )\). Give exact answers. Use fundamental identities as needed.

79. \(\sin^2 x(1-\sin^2 x)+\cos^2 x(1-\sin^2 x)=0\) 80. \(\sin^2 x-\cos^2 x-\sin x=0\) 81. \(\sin^2 x-\cos^2 x-1=0\) 82. \(\sin^2 x-\cos^2 x-\cos x=1\) 83. \(2\cos^2 x+3\sin x-3=0\) 84. \(\cos^2 x-2\sin x-2=0\) 85. \(2\sin^2 x-\cos x-1=0\) 86. \(2\sin^2 x-\cos x-2=0\) 87. \(4\sin \left(x\right)\cos \left(x\right)+2\sin \left(x\right)-2\cos \left(x\right)-1=0\)  88. \(2\sin \left(x\right)\cos \left(x\right)-\sin \left(x\right)+2\cos \left(x\right)-1=0\) 89. \(\sec \left(x\right)\sin \left(x\right)-2\sin \left(x\right)= 0\)

90. \(\dfrac{1}{\sec ^2 x}+2+\sin ^2 x+4\cos ^2 x=4\) 91. \( \cos (x) = \sin(-x) \) 92. \( \tan (x)\sin (x)  = 3\cos(x) \) 93. \( \tan (x) = \cot(x) \) 94. \(\tan^2(x)=-1+2\tan(-x)\) 95. \(\tan \left(x\right)\sin \left(x\right)-\sin \left(x\right)=0\) 96. \(\tan(x)-2\sin(x)\tan(x)=0\) 97. \(2\tan ^{2} \left(t\right)=3\sec \left(t\right)\) 98. \( \sec x = \tan x - 1\) 99. \(  \csc x = 1 - \cot x \)

Answers to odd exercises

79. \(\dfrac{\pi }{2}, \dfrac{3\pi }{2}\) 81. \(\dfrac{\pi}{2}\), \(\dfrac{3\pi}{2}\) 83. \(\dfrac{\pi}{6}\), \(\dfrac{5\pi}{6}\), \(\dfrac{\pi}{2}\)	85. \(\dfrac{\pi}{3}\), \(\pi\), \(\dfrac{5\pi}{3}\) 87. \(\dfrac{\pi}{6}\), \(\dfrac{2\pi}{3}\), \(\dfrac{5\pi}{6}\), \(\dfrac{4\pi}{3}\) 89. 0, \(\pi\), \(\dfrac{\pi}{3}\), \(\dfrac{5\pi}{3}\)	91.  \(\dfrac{3\pi}{4}\),  \(\dfrac{7\pi}{4}\)   93.  \(\dfrac{\pi}{4}\),  \(\dfrac{3\pi}{4}\), \(\dfrac{5\pi}{4}\), \(\dfrac{7\pi}{4}\)  95.  0, \(\pi\), \(\dfrac{\pi}{4}\),  \(\dfrac{5\pi}{4}\)	97. \(\dfrac{\pi}{3}\), \(\dfrac{5\pi}{3}\) 99. \(\dfrac{\pi}{2}\)

D: Use a calculator to solve linear equations

Exercise \(\PageIndex{D}\)

\( \bigstar \) Find all solutions on the interval \(0\le x<2\pi\).

101. \(\sin \left(x\right)=0.27\) 102. \(\sin \left(x\right)= -0.48\)

103. \(\sin \left(x\right)= -0.58\) 104. \(\cos \left(x\right)=-0.34\)

105. \(\cos \left(x\right)=-0.55\) 106. \(\cos \left(x\right)= 0.28\)

107. \(\tan \left(x\right)= 0.71\) 108. \(\tan \left(x\right)=-4.73\)

\( \bigstar \) Find the first two positive solutions

111. \(\csc \left(2x\right)-9=0\) 112. \(\sec \left(2\theta \right)=3\)

115. \(7\sin \left(6x\right)=2\) 116. \(7\sin \left(5x\right)= 6\) 117. \(3\sin \left(\dfrac{\pi }{4} x\right)=2\) 118. \(7\sin \left(\dfrac{\pi }{5} x\right)=6\)

119. \(5\cos \left(3x\right)=-3\) 120. \(3\cos \left(4x\right)=2\) 121. \(5\cos \left(\dfrac{\pi }{3} x\right)=1\) 122. \(3\cos \left(\dfrac{\pi }{2} x\right)=-2\)

\( \bigstar \) Use a calculator to find all solutions to four decimal places.

131. \(\tan x=-0.34\)

132. \(\sin x=-0.55\)

133. \(3\cos \left(\dfrac{\pi }{5} x\right)=2\)

134. \(8\cos \left(\dfrac{\pi }{2} x\right)=6\)

135. \(7\sin \left(3t\right)=-2\)

136. \(4\sin \left(4t\right)=1\)

Answers to odd exercises.

101.  \(0.2734,\; 2.8682\) \( \quad \)  103.  \(3.7603,\; 5.6645\) \( \quad \)  105.  \(2.1532,\; 4.1300\) \( \quad \)  107.  \(0.6174,\; 3.7590\)       

111.  \(0.056,\; 1.515,\; 3.197,\; 4.657\) \( \quad \)  115.  \(0.04829,\; 0.47531\)     
117.  \(0.9291,\; 3.0709\) \( \quad \)  119.  \(0.7381,\; 1.3563\) \( \quad \)  121.  \(1.3077,\; 4.6923\)

131.  \(\pi k-0.3277\) \( \quad \)  133.  \(0.1339 + 10k\)  and \(8.6614 + 10k\), where \(k\) is an integer     
135.  \(1.1438 + \dfrac{2\pi}{3} k\)  and \(1.9978 + \dfrac{2\pi}{3} k\), where \(k\) is an integer

E: Use calculator to Solve Quadratic Equations 

Exercise \(\PageIndex{E}\)

\( \bigstar \) Find all solutions on the interval \([0, 2\pi )\). Use the quadratic formula if the equations do not factor. 

147. \(10\sin \left(x\right)\cos \left(x\right)=6\cos \left(x\right)\) 148. \(-3\sin \left(t\right)=15\cos \left(t\right)\sin \left(t\right)\) 149. \(\sec ^{2} x=7\) 150. \(\csc ^{2} t=3\) 151. \(4\cos ^{2} (x)-4=15\cos \left(x\right)\) 152. \(8\sin ^{2} x+6\sin \left(x\right)+1=0\) 153. \(\sin^2 x-2\sin x-4=0\) 154. \(9\sin \left(w\right)-2=4\sin ^{2} (w)\)

155. \(\tan^2 x+3\tan x-3=0\) 156. \(6\sin^2 x-5\sin x+1=0\) 157. \(2\tan^2 x+9\tan x-6=0\) 158. \(6\tan^2 x+13\tan x=-6\) 159. \(-\tan^2 x-\tan x-2=0\) 160. \(5\cos^2 x+3\cos x-1=0\) 161. \(2\cos^2 x-\cos x+15=0\) 162. \(5\sin^2 x+2\sin x-1=0\)

163. \(3\cos^2 x-3\cos x-2=0\) 164. \(100\tan^2x+20\tan x-3=0\) 165. \(\tan^2 x+5\tan x-1=0\) 166. \(20\sin^2 x-27\sin x+7=0\) 167. \(8\cos^2 x-2\cos x-1=0\) 168. \(130\tan^2 x+69\tan x-130=0\) 169. \(2\tan^2 x+7\tan x+6=0\)

Answers to odd exercises.

147.  \(\dfrac{\pi}{2}\), \(\dfrac{3\pi}{2}\), \(0.644, \; 2.498\)  \( \quad \)  149.  \(1.183, \; 1.958, \; 4.325, \; 5.100 \)

151.  \(1.823, \; 4.460\) \( \quad \)  153.  There are no solutions.  \( \quad \)  155.  \(0.6694, \; 1.8287, \; 3.8110, \; 4.9703\)     
157.  \(0.5326, \; 1.7648, \; 3.6742, \; 4.9064\)    \( \quad \)  159.  There are no solutions. \( \quad \)  161.  There are no solutions.      

163.  \(\cos^{-1}\left(   \frac{3-\sqrt{33}}{3} \right ) \approx \; 2.0459, \;\; 2\pi -\cos^{-1}\left( \frac{3-\sqrt{33}}{3} \right)  \approx 4.2373\)

165.  \(\tan^{-1}\left (\tfrac{\sqrt{29}-5 }{2} \right ), \pi +\tan^{-1}\left (\tfrac{ -\sqrt{29}-5 }{2} \right),\)   \(\pi +\tan^{-1}\left (\tfrac{\sqrt{29}-5}{2} \right ), 2\pi +\tan^{-1}\left (\tfrac{-\sqrt{29}-5}{2} \right )\)

167.  \(\tfrac{\pi }{3}, \cos^{-1}\left ( -\tfrac{1}{4} \right ), 2\pi -\cos^{-1}\left ( -\tfrac{1}{4} \right ), \tfrac{5\pi }{3}\)

169.  \(\pi +\tan^{-1}(-2), \pi +\tan^{-1}\left (-\tfrac{3}{2}\right ), 2\pi +\tan^{-1}(-2), 2\pi +\tan^{-1}\left (-\tfrac{3}{2} \right )\)

\( \bigstar \) Find all solutions on the interval \([0, 2\pi )\). Use identities. Use the quadratic formula if the equations do not factor. 

171. \(12\sin ^{2} \left(t\right)+\cos \left(t\right)-6=0\) 172. \(6\cos ^{2} \left(x\right)+7\sin \left(x\right)-8=0\) 173. \(\cos ^{2} \phi =-6\sin \phi\) 174. \(\sin ^{2} t=\cos t\) 175. \(\tan \left(x\right)-3\sin \left(x\right)= 0\) 176. \(3\cos \left(x\right)=\cot \left(x\right)\)

177. \(\sin^2 x+\cos^2 x=0\) 178. \(\dfrac{2\tan x}{2-\sec ^2 x}-\sin^2 x=\cos^2 x \) 179. \(\tan^2 x-\sec x=1\) 180. \(\sin^2 x-2\cos^2 x=0\) 181. \(12\sin^2 t+\cos t-6=0\)

182. \(1-2\tan \left(w\right)=\tan ^{2} \left(w\right)\) 183. \(\csc^2 x-3\csc x-4=0\) 184. \(3\sec^2 x+2+\sin^2 x+\cos^2 x \\       =\tan^2 x\) 185. \(2\cos^2 x-\sin^2 x-\cos x-5=0\) 186. \(\tan^2 x-1-\sec^3 x \cos x=0\)

Answers to odd exercises.

171.  \(2.301, \;  3.983, \;  0.723, \;  5.560\)  \( \quad \)  173.  \(3.305, \;  6.120\) \( \quad \)  175.  \(0, \; \pi, \; 1.231, \; 5.052\) \( \quad \)  177.  There are no solutions.
179.  \(1.0472,3.1416,5.2360\)  \( \quad \) 181.  \(\cos^{-1}\left ( \tfrac{3}{4} \right ), \cos^{-1}\left ( -\tfrac{2}{3} \right ), 2\pi -\cos^{-1}\left ( -\tfrac{2}{3} \right ), 2\pi -\cos^{-1}\left ( \tfrac{3}{4} \right )\)
183.  \(\sin^{-1}\left ( \tfrac{1}{4} \right ), \pi -\sin^{-1}\left ( \tfrac{1}{4} \right ), \tfrac{3\pi }{2}\) \( \quad \)  185.  There are no solutions.