6.1e: Exercises - Inverse Trigonometric Functions

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A: Concepts.

Exercise $\PageIndex{A}$

Why do the functions $f(x)=\sin^{-1}x$ and $g(x)=\cos^{-1}x$ have different ranges?
Since the functions $y=\cos x$ and $y=\cos^{-1}x$ are inverse functions, why is $\cos^{-1}\left (\cos \left (-\dfrac{\pi }{6} \right ) \right )$not equal to $-\dfrac{\pi }{6}$?
Explain the meaning of $\dfrac{\pi }{6}=\arcsin (0.5)$.
Most calculators do not have a key to evaluate $\sec ^{-1}(2)$Explain how this can be done using the cosine function or the inverse cosine function.
Why must the domain of the sine function, $\sin x$be restricted to $\left [ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right ]$ for the inverse sine function to exist?
Discuss why this statement is incorrect: $\arccos(\cos x)=x$ for all $x$.
Determine whether the following statement is true or false and explain your answer: $\arccos(-x)=\pi - \arccos x$

Answers to odd exercises.

1. The function $y=\sin x$ is one-to-one on $\left [ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right ]$ ; thus, this interval is the range of the inverse function of $y=\sin x$, $f(x)=\sin^{-1}x$ The function $y=\cos x$ is one-to-one on $[0,\pi ]$ ; thus, this interval is the range of the inverse function of $y=\cos x$, $f(x)=\cos^{-1}x$

3. $\dfrac{\pi }{6}$ is the radian measure of an angle between $-\dfrac{\pi }{2}$ and $\dfrac{\pi }{2}$ whose sine is $0.5$.

5. In order for any function to have an inverse, the function must be one-to-one and must pass the horizontal line test. The regular sine function is not one-to-one unless its domain is restricted in some way. Mathematicians have agreed to restrict the sine function to the interval $\left [ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right ]$ so that it is one-to-one and possesses an inverse.

7. True . The angle, $\theta _1$ that equals $\arccos(-x)$, $x>0$, will be a second quadrant angle with reference angle, $\theta _2$, where $\theta _2$ equals $\arccos x$, $x>0$. Since $\theta _2$ is the reference angle for $\theta _1$, $\theta _2=\pi - \theta _1$ and $\arccos(-x)=\pi - \arccos x-$add texts here. Do not delete this text first.

B: Evaluate Inverse Trigonometric Functions for "Special Angles"

Exercise $\PageIndex{B}$

$ \bigstar $ Evaluate the expressions.

9. $\sin^{-1}(0)$

10. $\cos^{-1}(0)$

11. $\tan^{-1}(0)$

12. $\sin^{-1}(1)$

13. $\cos^{-1}(1)$

14. $\tan^{-1}(1)$

15. $\sin^{-1}\left(\dfrac{1}{2}\right)$

16. $\cos^{-1}\left(\dfrac{1}{2}\right)$

17. $\tan^{-1}\left(\dfrac{-1}{\sqrt{3}}\right)$

18. $\sin^{-1}\left(\dfrac{\sqrt{2}}{2}\right)$

19. $\cos^{-1}\left(\dfrac{\sqrt{2}}{2}\right)$

20. $\sin^{-1}\left(\dfrac{\sqrt{3}}{2}\right)$

21. $\cos^{-1}\left(\dfrac{\sqrt{3}}{2}\right)$

22. $\tan^{-1}(\sqrt{3})$

23. $\sin^{-1}(-1)$

24. $\cos^{-1}(-1)$

25. $\tan^{-1}(-1)$

26. $\sin^{-1}\left(-\dfrac{1}{2}\right)$

27. $\cos^{-1}\left(-\dfrac{1}{2}\right)$

28. $\tan^{-1}\left(-\dfrac{1}{\sqrt{3}}\right)$

29. $\cos^{-1}\left(-\dfrac{\sqrt{2}}{2}\right)$

30. $\sin^{-1}\left(-\dfrac{\sqrt{2}}{2}\right)$

31. $\sin^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)$

32. $\cos^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)$

33. $\tan^{-1}(-\sqrt{3})$

34. $\csc^{-1}( -2 )$

35. $\cot^{-1}(-\sqrt{3})$

36. $\sec^{1} \left( \dfrac{-2\sqrt{3}}{3} \right) $

Answers to odd exercises.: 9. $ 0$ 11. $ 0$ 13. $ 0$ 15. $\dfrac{\pi }{6}$ 17. $\dfrac{\pi }{6}$ 19. $\dfrac{\pi }{4}$ 21. $\dfrac{\pi }{6}$
23. $-\dfrac{\pi }{2}$ 25. $-\dfrac{\pi }{4}$ 27. $\dfrac{2\pi }{3}$ 29. $\dfrac{3\pi }{4}$ 31. $ - \dfrac{\pi }{3}$ 33. $-\dfrac{\pi }{3}$ 35. $ \tan^{-1} \left(\dfrac{-1}{\sqrt{3}}\right) = - \dfrac{\pi}{6}$

C: Evaluate Inverse Trigonometric Functions with a Calculator

Exercise $\PageIndex{C}$

$ \bigstar $ Use a calculator to evaluate each expression. Express answers to the nearest hundredth.

37. $\cos^{-1}(-0.4)$

38. $\arcsin (0.23)$

39. $\tan^{-1}(6)$

40 $ \sec^{-1} (2.5) $

41. $ \csc^{-1} (1.5) $

42. $ \cot^{-1} (5) $

Answers to odd exercises.: 37. $1.98$ 39. $1.41$ 41. $0.73$

D: Evaluate $ f^{-1} (f( \theta )) $ Compositions

Exercise $\PageIndex{D}$

$ \bigstar $ Evaluate without a calculator.

45. $\sin^{-1}\left(\sin \left( \dfrac{\pi}{10} \right)\right)$

46. $\sin^{-1}\left(\sin \left( \dfrac{2\pi}{3} \right)\right)$

47. $\sin^{-1}\left(\sin \left( \dfrac{5\pi}{4} \right)\right)$

48. $\sin^{-1}\left(\sin \left( -\dfrac{9\pi}{8} \right)\right)$

49. $\sin^{-1}\left(\sin \left( -\dfrac{3\pi}{2} \right)\right)$

50. $\sin^{-1}\left(\sin \left( -\dfrac{\pi}{7} \right)\right)$

51. $\cos^{-1}\left( \cos \left( \dfrac{\pi}{10} \right)\right)$

52. $\cos^{-1}\left( \cos \left( \dfrac{17\pi}{9} \right)\right)$

53. $\cos^{-1}\left( \cos \left( \dfrac{2\pi}{3} \right)\right)$

54. $\cos^{-1}\left( \cos \left( -\dfrac{11\pi}{6} \right)\right)$

55. $\cos^{-1}\left( \cos \left( -\dfrac{4\pi}{5} \right)\right)$

56. $\cos^{-1}\left( \cos \left( -\dfrac{\pi}{7} \right)\right)$

57. $\tan^{-1}\left(\tan \left( \dfrac{2\pi}{3} \right)\right)$

58. $\tan^{-1}\left(\tan \left( \dfrac{3\pi}{2} \right)\right)$

59. $\tan^{-1}\left(\tan \left( \dfrac{17\pi}{9} \right)\right)$

60. $\tan^{-1}\left(\tan \left( -\dfrac{11\pi}{6} \right)\right)$

61. $\tan^{-1}\left(\tan \left( -\dfrac{9\pi}{8} \right)\right)$

62. $\tan^{-1}\left(\tan \left( -\dfrac{4\pi}{5} \right)\right)$

Answers to odd exercises.: 45. $ \dfrac{\pi}{10} $ 47. $ - \dfrac{\pi}{4} $ 49. $ \dfrac{\pi}{2} $ 51. $ \dfrac{\pi}{10} $ 53. $ \dfrac{2\pi}{3} $ 55. $ \dfrac{4\pi}{5} $ 57. $ - \dfrac{\pi}{3} $ 59. $ - \dfrac{\pi}{9} $ 61. $ - \dfrac{\pi}{8} $

E: Evaluate $ f (f^{-1}( \frac{a}{b} )) $ Compositions

Exercise $\PageIndex{E}$

$ \bigstar $ Evaluate without a calculator.

65. $\sin \left(\sin^{-1} \left( \dfrac{1}{5} \right)\right)$

66. $\cos \left(\cos^{-1} \left( \dfrac{2}{3} \right)\right)$

67. $\tan \left(\tan^{-1} \left( 4 \right) \right) \\[6pt]$

68. $\sin \left(\sin^{-1} (-5) \right)$

69. $\cos \left(\cos^{-1} \left( -\dfrac{3}{2} \right)\right)$

70. $\tan \left(\tan^{-1} \left( -\dfrac{1}{4} \right)\right)$

Answers to odd exercises.: 65. $\dfrac{1}{5}$ $\qquad$ 67. $ 4 $ $\qquad$ 69. undefined

F: Evaluate $ f (g^{-1}( \frac{a}{b} )) $ Compositions

Exercise $\PageIndex{F}$

$ \bigstar $ Evaluate without a calculator.

71. $\sin \left(\cos^{-1} \left( -\dfrac{2}{\sqrt{5}} \right)\right)$

72. $\sin \left(\cos^{-1} \left( \dfrac{3}{4} \right)\right)$

73. $\sin \left(\cos^{-1} \left(\dfrac{3}{5} \right)\right)$

74. $\sin \left(\tan^{-1} \left(\dfrac{4}{3} \right)\right)$

75. $\sin \left(\tan^{-1} \left( -\dfrac{2}{\sqrt{5}} \right)\right)$

76. $\sin \left(\tan^{-1} \left( 1 \right)\right)$

77. $\cos \left(\sin^{-1} \left( \dfrac{1}{2} \right)\right)$

78. $\cos \left(\sin^{-1} \left(\dfrac{4}{5} \right)\right)$

79. $\cos \left(\tan^{-1} \left(\dfrac{12}{5} \right)\right)$

80. $\cos \left(\tan^{-1} \left( -\dfrac{1}{2} \right)\right)$

81. $\tan\left(\sin^{-1} \left( -\dfrac{2}{\sqrt{5}} \right)\right)$

82. $\tan\left(\sin^{-1} \left( -\dfrac{\sqrt{2}}{3} \right)\right)$

83. $\tan \left(\cos^{-1} \left( -\dfrac{\sqrt{3}}{2} \right)\right)$

84. $\cot \left(\cos^{-1} \left( \dfrac{2}{\sqrt{5}} \right)\right)$

85. $\cot \left(\sin^{-1} \left( \dfrac{\sqrt{3}}{2} \right)\right)$

86. $\sec \left(\tan^{-1} \left( -\dfrac{3}{4} \right)\right)$

87. $\sec\left(\sin^{-1} \left( -\dfrac{3}{4} \right)\right)$

88. $\csc \left(\tan^{-1} \left( \dfrac{\sqrt{2}}{3} \right)\right)$

89. $\csc \left(\cos^{-1} \left( \dfrac{\sqrt{2}}{3} \right)\right)$

90. Find the exact value of $\dfrac{\sin^{-1}\left ( \tfrac{1}{2} \right )-\cos^{-1}\left ( \tfrac{\sqrt{2}}{2} \right )+\sin^{-1}\left ( \tfrac{\sqrt{3}}{2} \right )-\cos^{-1}(1)}{\cos^{-1}\left ( \tfrac{\sqrt{3}}{2} \right )-\sin^{-1}\left ( \tfrac{\sqrt{2}}{2} \right )+\cos^{-1}\left ( \tfrac{1}{2} \right )-\sin^{-1}(0)}$

Answers to odd exercises.: 71. $\dfrac{\sqrt{5}}{5}$ 73. $\dfrac{4}{5}$ 75. $- \dfrac{2}{3}$ 77. $\dfrac{\sqrt{3}}{2}$ 79. $\dfrac{5}{13}$
81. $ - 2 $ 83. $ - \dfrac{\sqrt{3}}{3}$ 85. $ \dfrac{\sqrt{3}}{3}$ 87. $\dfrac{4\sqrt{7}}{7}$ 89. $\dfrac{3\sqrt{7}}{7}$

G: Evaluate $ f^{-1}(g( \theta )) $ Compositions

Exercise $\PageIndex{G}$

$ \bigstar $ Find the angle $\theta$ in the given right triangle. Round answers to the nearest hundredth.

102.
$CNX_Precalc_Figure_06_03_201.jpg$

103.
$CNX_Precalc_Figure_06_03_202.jpg$

$ \bigstar $ Find the exact value, if possible, without a calculator, or round to the nearest hundredth.

104. $\sin^{-1}\left(\cos \left( \dfrac{2\pi}{3} \right)\right)$

105. $\sin^{-1}\left(\cos \left(\dfrac{-\pi}{2} \right)\right)$

106. $\sin^{-1}(\cos(\pi))$

107. $\sin^{-1}\left(\tan \left( - \dfrac{4\pi}{3} \right)\right)$

108. $\cos^{-1}\left(\sin \left(\dfrac{\pi}{3} \right)\right)$

109. $\cos^{-1}\left(\sin \left( \dfrac{7\pi}{4} \right)\right)$

110. $\cos^{-1} \left( \sin \left(\dfrac{5\pi}{6} \right) \right) $

111. $\cos^{-1}\left(\cot \left( -\dfrac{3\pi}{4} \right)\right)$

112. $\tan^{-1}\left(\sin \left(\dfrac{4\pi}{3} \right)\right)$

113. $\tan^{-1}\left(\sin \left(\dfrac{\pi}{3} \right)\right)$

114. $\tan^{-1}(\sin(\pi))$

115. $\tan^{-1}\left(\sin \left(-\dfrac{5\pi}{2} \right)\right)$

116. $\tan^{-1}\left( \csc \left( \dfrac{7\pi}{6} \right)\right)$

117. $\tan^{-1}\left( \sec \left( -\dfrac{\pi}{6} \right)\right)$

Answers to odd exercises.: 103. $0.56$ radians 105. $0$ 107. undefined 109. $ \dfrac{3\pi}{4} $
111. $0$ 113. $0.71$ 115. $-\dfrac{\pi}{4}$ 117. $ 0.86 $

H: Evaluate $ f (g^{-1}( h(u) )) $ Compositions

Exercise $\PageIndex{H}$

For the exercises below, (a) Find the exact value of the expression in terms of $u$. (b) State any restrictions to $u$.

121. $\cos \left( \sin^{-1} \left( u\right)\right)$

122. $\tan \left( \sin^{-1} \left( u\right)\right)$

123. $\sin \left( \tan^{-1} \left( u\right)\right)$

124. $\cos \left( \tan^{-1} \left( u\right)\right)$

125. $\tan \left( \cos^{-1} \left( u\right)\right)$

126. $\sin \left( \cos^{-1} \left( u\right)\right)$

127. $\tan \left(\sin^{-1} (u-1)\right)$

128. $\cos \left(\sin^{-1} (1-u)\right)$

129. $\cos \left(\sin^{-1} \left(\dfrac{1}{u}\right)\right)$

130. $\tan \left(\sin^{-1} \left(\dfrac{u}{u+1}\right)\right)$

131. $\sin \left(\tan^{-1} \left(u+\dfrac{1}{2}\right)\right)$

132. $\cos \left(\tan^{-1} (3u-1)\right)$

133. $ \sin \left( \tan^{-1} \left(\dfrac{u}{\sqrt{2u+1}}\right) \right)$

Answers to odd exercises.: 121. $ \sqrt{1-u^2} $, $ -1 \le u \le 1 $    123. $ \dfrac{u}{\sqrt{1+u^2} } $, no restrictions   125. $ \dfrac{\sqrt{1-u^2}}{u} $, $ -1 \le u \le 1 $
127. $\dfrac{u-1}{\sqrt{-u^2+2u}}$, $ 0 \le u \le 2 $     129. $\dfrac{\sqrt{u^2-1}}{u}$, $ u \ge 1 $ or $ u \le -1 $ 131. $\dfrac{u+\tfrac{1}{2}}{\sqrt{u^2+u+\tfrac{5}{4}}}$, no restrictions
133. $\dfrac{u}{u+1}$, $ u \gt -\dfrac{1}{2} $

I: Graphs of inverses

Exercise $\PageIndex{I}$

138. Graph $y=\sin^{-1} x$ and state the domain and range of the function.

139. Graph $y=\arccos x$ and state the domain and range of the function.

140. Graph one cycle of $y=\tan^{-1} x$ and state the domain and range of the function.

Answers to odd exercises.: 139. $Ex 6.3.49.png$ domain $[-1,1]$range $[0,\pi ]$

J: Applications

Exercise $\PageIndex{J}$

143. Suppose a $13$-foot ladder is leaning against a building, reaching to the bottom of a second-ﬂoor window $12$ feet above the ground. What angle, in radians, does the ladder make with the building?

144. Suppose you drive $0.6$ miles on a road so that the vertical distance changes from $0$ to $150$ feet. What is the angle of elevation of the road?

145. An isosceles triangle has two congruent sides of length $9$ inches. The remaining side has a length of $8$ inches. Find the angle that a side of $9$ inches makes with the $8$-inch side.

146. Without using a calculator, approximate the value of $\arctan (10,000)$Explain why your answer is reasonable.

147. A truss for the roof of a house is constructed from two identical right triangles. Each has a base of $12$ feet and height of $4$ feet. Find the measure of the acute angle adjacent to the $4$-foot side.

148. The line $y=\dfrac{3}{5}x$ passes through the origin in the $x,y$-plane. What is the measure of the angle that the line makes with the positive $x$-axis?

149. The line $y=\dfrac{-3}{7}x$ passes through the origin in the $x,y$-plane. What is the measure of the angle that the line makes with the negative $x$-axis?

150. What percentage grade should a road have if the angle of elevation of the road is $4$ degrees? (The percentage grade is defined as the change in the altitude of the road over a $100$-foot horizontal distance. For example a $5\%$ grade means that the road rises $5$ feet for every $100$ feet of horizontal distance.)

151. A $20$-foot ladder leans up against the side of a building so that the foot of the ladder is $10$ feet from the base of the building. If specifications call for the ladder's angle of elevation to be between $35$ and $45$ degrees, does the placement of this ladder satisfy safety specifications?

152. Suppose a $15$-foot ladder leans against the side of a house so that the angle of elevation of the ladder is $42$ degrees. How far is the foot of the ladder from the side of the house?

Answers to odd exercises.: 143. $0.395$ radians 145. $1.11$ radians 147. $1.25$ radians 149. $0.405$ radians
151. No. The angle the ladder makes with the horizontal is $60$ degrees..

A: Concepts.

B: Evaluate Inverse Trigonometric Functions for "Special Angles"

C: Evaluate Inverse Trigonometric Functions with a Calculator

D: Evaluate \( f^{-1} (f( \theta )) \) Compositions

E: Evaluate \( f (f^{-1}( \frac{a}{b} )) \) Compositions

F: Evaluate \( f (g^{-1}( \frac{a}{b} )) \) Compositions

G: Evaluate \( f^{-1}(g( \theta )) \) Compositions

H: Evaluate \( f (g^{-1}( h(u) )) \) Compositions

I: Graphs of inverses

J: Applications