Chapter 8: More Functions and Identities
Exercises: 8.2 Inverse Trigonometric Functions Exercises
SKILLS
Practice each skill in the Homework Problems listed.
- Decide whether a function has an inverse function #1–8
- Evaluate the inverse trig functions #9–20
- Model problems with inverse trig functions #21–24
- Solve formulas #25–30
- Simplify expressions involving the inverse trig functions #31–42, 51–68
- Graph the inverse trig functions #43–50, 69 and 70
Suggested Homework
Exercises for 8.2 Inverse Trigonometric Functions
Exercise Group
In Problems 1–4, which functions have an inverse function? Explain your answer.
2.
3.
4.
Exercise Group
For Problems 5–8, graph the function and decide if it has an inverse function.
5.
[latex]f(x)=\sin 2x - \cos x[/latex]
6.
[latex]g(x)=4e^{-(x/4)^2}[/latex]
7.
[latex]G(x)=\sqrt{25-x^2}[/latex]
8.
[latex]F(x)=\ln(x^3+8)[/latex]
Exercise Group
For Problems 9–14, use a calculator to evaluate. Round your answers to the nearest tenth of a degree.
9.
[latex]\sin^{-1}(0.2838)[/latex]
10.
[latex]\tan^{-1}(4.8972)[/latex]
11.
[latex]\cos^{-1}(0.6894)[/latex]
12.
[latex]\arccos(-0.8134)[/latex]
13.
[latex]\arctan(-1.2765)[/latex]
14.
[latex]\arcsin(-07493)[/latex]
Exercise Group
For Problems 15–20, give exact values in radians.
15.
[latex]\cos^{-1} \dfrac{-1}{\sqrt{2}}[/latex]
16.
[latex]\tan^{-1}(-1)[/latex]
17.
[latex]\sin^{-1}\dfrac{-1}{2}[/latex]
18.
[latex]\arccos\dfrac{\sqrt{3}}{2}[/latex]
19.
[latex]\arctan \dfrac{1}{\sqrt{3}}[/latex]
20.
[latex]\arcsin (-1)[/latex]
Exercise Group
For Problems 21–26, sketch a figure to help you model each problem.
21.
Delbert is watching the launch of a satellite at Cape Canaveral. The viewing area is 500 yards from the launch site. The angle of elevation, [latex]\theta{,}[/latex] of Delbert’s line of sight increases as the booster rocket rises.
- Write a formula for the height, [latex]h{,}[/latex] of the rocket as a function of [latex]\theta{.}[/latex]
- Write a formula for [latex]\theta[/latex] as a function of [latex]h{.}[/latex]
- Evaluate the formula in part (b) for [latex]h=1000{,}[/latex] and interpret the result.
22.
Francine’s house lies under the flight path from the city airport, and commercial airliners pass overhead at an altitude of 35,000 feet. As Francine watches an airplane recede, its angle of elevation, [latex]\theta{,}[/latex] decreases.
- Write a formula for the horizontal distance, [latex]d{,}[/latex] to the airplane as a function of [latex]\theta{.}[/latex]
- Write a formula for [latex]\theta[/latex] as a function of [latex]d{.}[/latex]
- Evaluate the formula in part (b) for [latex]d=20,000{,}[/latex] and interpret the result.
23.
While driving along the interstate, you approach an enormous 50-foot-wide billboard that sits just beside the road. Your viewing angle, [latex]\theta{,}[/latex] increases as you get closer to the billboard.
- Write a formula for your distance, [latex]d{,}[/latex] from the billboard as a function of [latex]\theta{.}[/latex]
- Write a formula for [latex]\theta[/latex] as a function of [latex]d{.}[/latex]
- Evaluate the formula in part (b) for [latex]d=200{,}[/latex] and interpret the result.
24.
Emma is walking along the bank of a straight river toward a 20-meter-long bridge over the river. Let [latex]\theta[/latex] be the angle subtended horizontally by Emma’s view of the bridge.
- Write a formula for Emma’s distance from the bridge, [latex]d{,}[/latex] as a function of [latex]\theta{.}[/latex]
- Write a formula for [latex]\theta[/latex] as a function of [latex]d{.}[/latex]
- Evaluate the formula in part (b) for [latex]d=500{,}[/latex] and interpret the result.
25.
Martin is viewing a 4-meter-tall painting whose base is 1 meter above his eye level.
- Write a formula for [latex]\alpha{,}[/latex] the angle subtended from Martin’s eye level to the bottom of the painting, when he stands [latex]x[/latex] meters from the wall.
- Write a formula for [latex]\beta{,}[/latex] the angle subtended by the painting, in terms of [latex]x{.}[/latex]
- Evaluate the formula in part (b) for [latex]x=5{,}[/latex] and interpret the result.
26.
A 5-foot mirror is positioned so that its bottom is 1.5 feet below Jane’s eye level.
- Write a formula for [latex]\alpha{,}[/latex] the angle subtended by the section of mirror below Jane’s eye level, when she stands [latex]x[/latex] feet from the mirror.
- Write a formula for [latex]\theta{,}[/latex] the angle subtended by the entire mirror, in terms of [latex]x{.}[/latex]
- Evaluate the formula in part (b) for [latex]x=10{,}[/latex] and interpret the result.
Exercise Group
For Problems 27–32, solve the formula for the given variable.
27.
[latex]V=V_0 \sin(2\pi\omega t+\phi){,}[/latex] for [latex]t[/latex]
28.
[latex]R=\dfrac{1}{32} v_0^2\sin (2\theta){,}[/latex] for [latex]\theta[/latex]
29.
[latex]\dfrac{a}{\sin A}= \dfrac{b}{\sin B}{,}[/latex] for [latex]A[/latex]
30.
[latex]c^2=a^2 + b^2 - 2ab\cos C{,}[/latex] for [latex]C[/latex]
31.
[latex]P=\dfrac{k}{R^4\cos \theta}[/latex] for [latex]\theta[/latex]
32.
[latex]\dfrac{r}{z}=\dfrac{1}{\tan (\alpha + \beta)}{,}[/latex] for [latex]\alpha[/latex]
Exercise Group
For Problems 33–38, find exact values without using a calculator.
33.
[latex]\tan\left(\sin^{-1}\left(\dfrac{2}{3}\right)\right)[/latex]
34.
[latex]\tan\left(\cos^{-1}\left(\dfrac{3}{4}\right)\right)[/latex]
35.
[latex]\cos\left[\tan^{-1}(-2)\right][/latex]
36.
[latex]\sin \left[\tan^{-1}\left(\dfrac{-3}{\sqrt{5}}\right)\right][/latex]
37.
[latex]\sin\left[\cos^{-1}\left(\dfrac{-2\sqrt{6}}{7}\right)\right][/latex]
38.
[latex]\cos\left[\sin^{-1}\left(\dfrac{-2}{7}\right)\right][/latex]
Exercise Group
For Problems 39–44, simplify the expression.
39.
[latex]\tan(\cos^{-1}x)[/latex]
40.
[latex]\cos(\tan^{-1}a)[/latex]
41.
[latex]\cos(\sin^{-1}h)[/latex]
42.
[latex]\sin(\cos^{-1}v)[/latex]
43.
[latex]\sin(\tan^{-1}2t)[/latex]
44.
[latex]\tan(\sin^{-1}3b)[/latex]
Exercise Group
For Problems 45–47, complete the table of values and sketch the function.
45.
| [latex]x[/latex] | [latex]-1[/latex] | [latex]\dfrac{-\sqrt{3}}{2}[/latex] | [latex]\dfrac{-\sqrt{2}}{2}[/latex] | [latex]\dfrac{-1}{2}[/latex] | [latex]0[/latex] | [latex]\dfrac{1}{2}[/latex] | [latex]\dfrac{\sqrt{2}}{2}[/latex] | [latex]\dfrac{\sqrt{3}}{2}[/latex] | [latex]1[/latex] |
| [latex]\cos^{-1}x[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
46.
| [latex]x[/latex] | [latex]-1[/latex] | [latex]\dfrac{-\sqrt{3}}{2}[/latex] | [latex]\dfrac{-\sqrt{2}}{2}[/latex] | [latex]\dfrac{-1}{2}[/latex] | [latex]0[/latex] | [latex]\dfrac{1}{2}[/latex] | [latex]\dfrac{\sqrt{2}}{2}[/latex] | [latex]\dfrac{\sqrt{3}}{2}[/latex] | [latex]1[/latex] |
| [latex]\sin^{-1}x[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
47.
| [latex]x[/latex] | [latex]-\sqrt{3}[/latex] | [latex]-1[/latex] | [latex]\dfrac{-1}{\sqrt{3}}[/latex] | [latex]0[/latex] | [latex]\dfrac{1}{\sqrt{3}}[/latex] | [latex]1[/latex] | [latex]\sqrt{3}[/latex] |
| [latex]\tan^{-1}x[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
48.
Use a graphing calculator to answer each of the following questions. Then explain the results.
- Does [latex]\cos^{-1}x=\dfrac{1}{\cos x}{?}[/latex]
- Does [latex]\sin^{-1}x=\dfrac{1}{\sin x}{?}[/latex]
- Does [latex]\tan^{-1}x=\dfrac{1}{\tan x}{?}[/latex]
49.
- Sketch a graph of [latex]y=\cos^{-1}x{,}[/latex] and label the scales on the axes.
- Use transformations to sketch graphs of [latex]y=2\cos^{-1}x[/latex] and [latex]y=\cos^{-1}(2x){.}[/latex]
Does [latex]2\cos^{-1}x=\cos^{-1}(2x){?}[/latex]
50.
- Sketch a graph of [latex]y=\sin^{-1}x{,}[/latex] and label the scales on the axes.
- Use transformations to sketch graphs of [latex]y=\dfrac{1}{2}\sin^{-1}x[/latex] and [latex]y=\cos^{-1}\left(\dfrac{1}{2}x\right){.}[/latex]
- Does [latex]\dfrac{1}{2}\sin^{-1}x=\cos^{-1}\left(\dfrac{1}{2}x\right){?}[/latex]
51.
- Sketch a graph of [latex]y=\tan^{-1}x{,}[/latex] and label the scales on the axes.
- Use your calculator to graph [latex]y=\dfrac{\sin^{-1}x}{\cos^{-1}x}[/latex] on a suitable domain.
- Does [latex]\tan^{-1}x=\dfrac{\sin^{-1}x}{\cos^{-1}x}{?}[/latex]
52.
- Use your calculator to sketch [latex]y=\sqrt[3]{x}[/latex] and [latex]\tan^{-1}x[/latex] on [latex][-10, 10]{.}[/latex]
- Describe the similarities and differences in the two graphs.
Exercise Group
Use the identities from Section 8.1 to help you find exact values for the expressions in Problems 53–58.
53.
[latex]\sin (2\tan^{-1}4)[/latex]
54.
[latex]\cos\left(2\sin^{-1}\left(\dfrac{5}{13}\right)\right)[/latex]
55.
[latex]\tan\left(2\cos^{-1}\left(\dfrac{3}{4}\right)\right)[/latex]
56.
[latex]\sin\left(2\cos^{-1}\left(\dfrac{-4}{5}\right)\right)[/latex]
57.
[latex]\tan\left(2\sin^{-1}\left(\dfrac{1}{3}\right)\right)[/latex]
58.
[latex]\cos\left(2\tan^{-1}\left(\dfrac{3}{2}\right)\right)[/latex]
59.
Let [latex]\alpha= \cos^{-1}\left(\dfrac{-4}{5}\right),~\beta=\sin^{-1}\left(\dfrac{5}{13}\right){.}[/latex] Find exact values for the following:
[latex]\displaystyle \sin (\alpha+\beta)[/latex]
[latex]\displaystyle \cos (\alpha-\beta)[/latex]
[latex]\displaystyle \sin (\alpha-\beta)[/latex]
60.
Let [latex]\alpha= \sin^{-1}\left(\dfrac{-15}{17}\right),~\beta=\tan^{-1}\left(\dfrac{4}{3}\right){.}[/latex] Find exact values for the following:
[latex]\displaystyle \sin (\alpha+\beta)[/latex]
[latex]\displaystyle \cos (\alpha-\beta)[/latex]
[latex]\displaystyle \sin (\alpha-\beta)[/latex]
61.
Find an exact value for [latex]\sin\left(\tan^{-1}\left(\dfrac{3}{4}\right)-\sin^{-1}\left(\dfrac{-4}{5}\right)\right){.}[/latex]
62.
Find an exact value for [latex]\cos\left(\tan^{-1}\left(\dfrac{5}{12}\right)+\sin^{-1}\left(\dfrac{-3}{5}\right)\right){.}[/latex]
63.
Express in terms of [latex]x[/latex] without trigonometric functions.
[latex]\displaystyle \cos(2\sin^{-1}x)[/latex]
64.
Express in terms of [latex]w[/latex] without trigonometric functions.
[latex]\displaystyle \cos(2\tan^{-1}w)[/latex]
65.
If [latex]x=5\sin \theta,~0° \lt \theta \lt 90°{,}[/latex] express [latex]\sin 2\theta[/latex] and [latex]\cos 2\theta[/latex] in terms of [latex]x{.}[/latex]
66.
If [latex]x-1=2\cos \theta,~0° \lt \theta \lt 90°{,}[/latex] express [latex]\sin 2\theta[/latex] and [latex]\cos 2\theta[/latex] in terms of [latex]x{.}[/latex]
67.
If [latex]x=3\tan \theta{,}[/latex] write [latex]\theta +\dfrac{1}{4}\sin 2\theta[/latex] in terms of [latex]x{.}[/latex]
68.
If [latex]x=5\cos \theta{,}[/latex] write [latex]\dfrac{\theta}{2}-\cos 2\theta[/latex] in terms of [latex]x{.}[/latex]
69.
- For what values of [latex]x[/latex] is the function [latex]f(x)=\sin (\arcsin x)[/latex] defined?
- Is [latex]\sin (\arcsin x)=x[/latex] for all [latex]x[/latex] where it is defined? If not, for what values of [latex]x[/latex] is the equation false?
- For what values of [latex]x[/latex] is the function [latex]g(x)=\arcsin(\sin x)[/latex] defined?
- Is [latex]\arcsin (\sin x)=x[/latex] for all [latex]x[/latex] where it is defined? If not, for what values of [latex]x[/latex] is the equation false?
70.
- For what values of [latex]x[/latex] is the function [latex]f(x)=\cos (\arccos x)[/latex] defined?
- Is [latex]\cos (\arccos x)=x[/latex] for all [latex]x[/latex] where it is defined? If not, for what values of [latex]x[/latex] is the equation false?
- For what values of [latex]x[/latex] is the function [latex]g(x)=\arccos(\cos x)[/latex] defined?
- Is [latex]\arccos (\cos x)=x[/latex] for all [latex]x[/latex] where it is defined? If not, for what values of [latex]x[/latex] is the equation false?
71.
Use your calculator to graph [latex]y=\sin^{-1}x+\cos^{-1}x{.}[/latex]
- State the domain and range of the graph.
- Explain why the graph looks as it does.
72.
Use your calculator to graph [latex]y=\tan^{-1}x+\tan^{-1}(\dfrac{1}{x}){.}[/latex]
- State the domain and range of the graph.
- Explain why the graph looks as it does.
Exercise Group
In Problems 73–74, we find a formula for the area under part of a semicircle.
73.
Use the figure of a unit circle to answer the following.
- Write an expression for the area of the shaded sector in terms of [latex]\theta{.}[/latex]
- How are [latex]\theta[/latex] and [latex]t[/latex] related in the figure? (Hint: Write an expression for [latex]\sin \theta{.}[/latex])
- Combine your answers to (a) and (b) to write an expression for the area of the sector in terms of [latex]t{.}[/latex]
74.
Use the figure of a unit circle to answer the following.
- Write an expression for the height of the shaded triangle in terms of [latex]t{.}[/latex] (Hint: Use the Pythagorean theorem.)
- Write an expression for the area of the triangle in terms of [latex]t{.}[/latex]
- Combine your answers to (b) and to Problem 73 to write an expression for the area bounded above by the upper semicircle, below by the [latex]x[/latex]-axis, on the left by the [latex]y[/latex]-axis, and on the right by [latex]x=t{,}[/latex] when [latex]0 \le t \le 1{.}[/latex]