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Reed College Department of Mathematics & Statistics

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Worksheet Inverse Trig Functions

2.

Suppose a 6-foot ladder is leaning against a building, reaching to the bottom of a second-floor window 5 feet above the ground. What angle, in degrees, does the ladder make with the building? Round to the nearest hundredth.
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Answer.
\(33.5573\)
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3.

Suppose you drive 0.5 miles on a road, during which the vertical distance changes from 0 to 340 feet. What is the angle of elevation (in degrees) of the road? Round to the nearest hundredth.
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Answer.
\(7.39955\)
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4.

Given that \(\>t = \tan ^{-1} (\frac {4} {3}) \>\text{,}\) find the exact value of
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(a) \(\> \sin (t)\text{:}\)
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(b) \(\> \sec (t)\text{:}\)
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Answer 1.
\(0.8\)
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Answer 2.
\(1.66667\)
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5.

Complete the identity using the triangle method
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(a) \(\cos (\tan ^{-1} (x))\) =
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(b) \(\sin ( \sec ^{-1} (x))\) =
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(The inverse functions are obtained by restricting \(\tan\) to the domain \(-\pi/2 \lt \theta \lt \pi/2\text{,}\) and \(\sec\) to the domain \([0,\pi/2) \cup (\pi/2,\pi]\text{,}\) and inverting them on these restricted domains. Your answers should work for every \(x\) where the inverse functions are defined. Take care with signs!)
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Answer 1.
\(\frac{1}{\sqrt{x^{2}+1}}\)
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Answer 2.
\(\sqrt{1-\frac{1}{x^{2}}}\)
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6.

Find the exact value of each expression if defined; otherwise, input undefined.
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(a) \(\tan(\sin^{-1} \frac{1}{2}) =\) .
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(b) \(\tan(\sin^{-1} (-\frac{1}{2})) =\) .
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Answer 1.
\(\frac{1}{\sqrt{3}}\)
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Answer 2.
\(\frac{-1}{\sqrt{3}}\)
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7.

Find the exact value of each expression if defined; otherwise, input undefined.
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(a) \(\cos(\sin^{-1} \frac{\sqrt{3}}{2}) =\) .
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(b) \(\sin(\cos^{-1} \frac{\sqrt{3}}{2}) =\) .
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Answer 1.
\(\frac{1}{2}\)
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Answer 2.
\(\frac{1}{2}\)
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8.

State the domain and range of each of the following functions:
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(Enter your answers as inequalities: . Your domains should be inequalities in \(x\) and ranges, in \(y\text{.}\) As always, type pi for \(\pi\text{.}\) If you wish to enter \(\infty\text{,}\) type infinity.)
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(a) \(f(x) = \arccos{(x)}\) has
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Domain:
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Range:
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(b) \(g(x) = \arcsin{(x)}\) has
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Domain:
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Range:
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(c) \(h(x) = \arctan{(x)}\) has
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Domain:
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Range:
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Answer 1.
\(-1\le x\le 1\)
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Answer 2.
\(0\le y\le \pi \)
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Answer 3.
\(-1\le x\le 1\)
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Answer 4.
\(\frac{-\pi }{2}\le y\le \frac{\pi }{2}\)
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Answer 5.
\(-\infty < x < \infty \)
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Answer 6.
\(\frac{-\pi }{2} < y < \frac{\pi }{2}\)
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Solution.
SOLUTION\(f\)\(-1 \le x \le 1\text{,}\)\(\arccos{(x)}\)\(x\)\(-1\)\(1\)\(0\)\(\pi\)\(g\)\(-1 \le x \le 1\text{,}\)\(\arcsin{(x)}\)\(x\)\(-1\)\(1\)\(-\frac{\pi}{2}\)\(\frac{\pi}{2}\)\(h\)\(\arctan{(x)}\)\(x\text{.}\)\(-\frac{\pi}{2}\)\(\frac{\pi}{2}\text{.}\)
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9.

Find the exact value of each expression:
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(a) \(\sec(\arctan(2))\)
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(b) \(\cos(2\sin^{-1}(\frac{5}{13}))\)
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(a)
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(b)
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Answer 1.
\(2.23607\)
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Answer 2.
\(0.704142\)
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10.

Find the domain and the range of \(g(x)=\sin^{-1}(3x+1)\text{.}\)
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Domain: \(\leq x \leq\)
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Range: \(\leq y \leq\)
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Answer 1.
\(-0.666666666666667\)
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Answer 2.
\(0\)
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Answer 3.
\(\frac{-\pi }{2}\)
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Answer 4.
\(\frac{\pi }{2}\)
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11.

Evaluate the following expressions.
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a. \(\sin \left( \sin^{-1} \left( -1 \right) \right) =\)
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b. \(\cos \left( \cos^{-1} \left( \frac{\sqrt{2}}{2} \right) \right) =\)
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c. \(\tan \left( \tan^{-1} \left( 1 \right) \right) =\)
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Answer 1.
\(-1\)
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Answer 2.
\(\frac{\sqrt{2}}{2}\)
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Answer 3.
\(1\)
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12.

Find the exact value.
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\(\small{\sin} \left(\small{2 \cos^{-1}} \large{\left(\frac{5}{13}\right)} \right)\) =
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Answer.
\(2\cdot 0.923077\cdot 0.384615\)
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Solution.
\(\small{^{-1}}\text{,}\)\(\small{angle}\text{.}\)\(\small{\theta}\text{,}\)\(\small{\sin(2 \theta)}\text{,}\)\(\small{\theta} = \large{\frac{5}{13}}\text{.}\)\(\small{(2\theta) = 2\mathrm{sin}\theta \mathrm{cos}\theta}\text{.}\)\(\small{\theta} = \large{\frac{12}{13}}\text{.}\)\(\small{2\theta = 2} \left( \large{\frac{12}{13}} \right) \left( \large{\frac{5}{13}} \right) \small{=} \large{\frac{120}{169} }\text{.}\)
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13.

Find the inverse of the following function and state its domain.
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\(f(x)=8 \sin (15 x)\)
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Type ’arcsin’ for the inverse sine function in your answer.
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\(\ f^{-1} (x)\) =
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Domain= [ , ]
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Answer 1.
\(0.0666667\sin^{-1}\mathopen{}\left(\frac{x}{8}\right)\)
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Answer 2.
\(-8\)
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Answer 3.
\(8\)
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14.

Find the inverse of the following function and state its domain.
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\(f(x)=10 \cos (11 x) + 2\)
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Type ’arccos’ for the inverse cosine function in your answer.
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\(\ f^{-1} (x)\) =
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Domain= [ , ]
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Answer 1.
\(0.0909091\cos^{-1}\mathopen{}\left(\frac{x-2}{10}\right)\)
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Answer 2.
\(-8\)
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Answer 3.
\(12\)
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15.

Find the value of each expression if defined; otherwise, input undefined.
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(a) \(\tan(\tan^{-1} 6) =\) .
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(b) \(\sin^{-1}[\sin (-\frac{\pi}{6})] =\) radians.
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Answer 1.
\(6\)
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Answer 2.
\(\frac{-\pi }{6}\)
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