1.
[latex]\sin^2 x \cos y[/latex]
Chapter 5: Equations and Identities
Exercises: Chapter 5 Review Problems
Suggested Problems
Problems: #4, 8, 10, 30, 34, 42, 50, 58, 62, 64, 68, 72, 76
Exercises for Chapter 5 Review
Exercise Group
For Problems 1–4, evaluate the expressions for [latex]x = 120°,~ y = 225°,[/latex] and [latex]z = 90°{.}[/latex] Give exact values for your answers.
2.
[latex]\sin z - \dfrac{1}{2} \sin y[/latex]
3.
[latex]\tan (z - x) \cos (y - z)[/latex]
4.
[latex]\dfrac{\tan^2 x}{2 \cos y}[/latex]
Exercise Group
For Problems 5–8, evaluate the expressions using a calculator. Are they equal?
5.
- [latex]\sin (20° + 40°)[/latex]
- [latex]\sin 20° + \sin 40°[/latex]
6.
- [latex]\cos^2 70° - \sin^2 70°[/latex]
- [latex]\cos (2\cdot 70°)[/latex]
7.
- [latex]\dfrac{\sin 55°}{\cos 55°}[/latex]
- [latex]\tan 55°[/latex]
8.
- [latex]\tan 80° - \tan 10°[/latex]
- [latex]\tan (80° - 10°)[/latex]
Exercise Group
For Problems 9–12, simplify the expression.
9.
[latex]3\sin x - 2\sin x \cos y + 2\sin x - \cos y[/latex]
10.
[latex]\cos t + 3\cos 3t - 3\cos t - 2\cos 3t[/latex]
11.
[latex]6 \tan^2 \theta + 2\tan \theta - (4\tan \theta )^2[/latex]
12.
[latex]\sin \theta (2\cos \theta - 2) + \sin \theta (1 - \sin \theta)[/latex]
Exercise Group
For Problems 13–16, decide whether or not the expressions are equivalent. Explain.
13.
[latex]\cos \theta + \cos 2\theta;~~\cos 3\theta[/latex]
14.
[latex]1 + \sin^2 x;~~(1 + \sin x)^2[/latex]
15.
[latex]3\tan^2 t - \tan^2 t;~~2\tan^2 t[/latex]
16.
[latex]\cos 4\theta;~~2\cos 2\theta[/latex]
Exercise Group
For Problems 17–20, multiply or expand.
17.
[latex](\cos \alpha + 2)(2\cos \alpha - 3)[/latex]
18.
[latex](1 - 3\tan \beta)^2[/latex]
19.
[latex](\tan \phi - \cos \phi)^2 = 0[/latex]
20.
[latex](\sin \rho - 2\cos \rho)(\sin \rho + \cos \rho)[/latex]
Exercise Group
For Problems 21–24, factor the expression.
21.
[latex]12\sin 3x - 6\sin 2x[/latex]
22.
[latex]2\cos^2 \beta + \cos \beta[/latex]
23.
[latex]1 - 9\tan^2 \theta[/latex]
24.
[latex]\sin^2 \phi - \sin \phi \tan \phi - 2\tan^2 \phi[/latex]
Exercise Group
For Problems 25–30, reduce the fraction.
25.
[latex]\dfrac{\cos^2 \alpha - \sin^2 \alpha}{\cos \alpha - \sin \alpha}[/latex]
26.
[latex]\dfrac{1 - \tan^2 \theta}{1 - \tan \theta}[/latex]
27.
[latex]\dfrac{3\cos x + 9}{2\cos x + 6}[/latex]
28.
[latex]\dfrac{5\sin \theta - 10}{\sin^2 \theta - 4}[/latex]
29.
[latex]\dfrac{3\tan^2 C - 12}{\tan^2 C - 4\tan C + 4}[/latex]
30.
[latex]\dfrac{\tan^2 \beta - \tan \beta - 6}{\tan \beta - 3}[/latex]
Exercise Group
For Problems 31–32, use a graph to solve the equation for [latex]0° \le x \lt 360°{.}[/latex] Check your solutions by substitution.
31.
[latex]8\cos x - 3 = 2[/latex]
32.
[latex]6\tan x - 2 = 8[/latex]
Exercise Group
For Problems 33–40, find all solutions between [latex]0°[/latex] and [latex]360°{.}[/latex] Give exact answers.
33.
[latex]2\cos^2 \theta + \cos \theta = 0[/latex]
34.
[latex]\sin^2 \alpha - \sin \alpha = 0[/latex]
35.
[latex]2\sin^2 x - \sin x - 1 = 0[/latex]
36.
[latex]\cos^2 B + 2\cos B + 1 = 0[/latex]
37.
[latex]\tan^2 x = \dfrac{1}{3}[/latex]
38.
[latex]\tan^2 t - \tan t = 0[/latex]
39.
[latex]6\cos^2 \alpha - 3\cos \alpha - 3 = 0[/latex]
40.
[latex]2\sin^2 \theta + 4\sin \theta + 2 = 0[/latex]
Exercise Group
For Problems 41–44, solve the equation for [latex]0° \le x \lt 360°{.}[/latex] Round your answers to two decimal places.
41.
[latex]2 - 5\tan \theta = -6[/latex]
42.
[latex]3 + 5\cos \theta = 4[/latex]
43.
[latex]3\cos^2 x + 7\cos x = 0[/latex]
44.
[latex]8 - 9\sin^2 x = 0[/latex]
45.
A light ray passes from glass to water, with a [latex]37°[/latex] angle of incidence. What is the angle of refraction? The index of refraction from water to glass is 1.1.
46.
A light ray passes from glass to water, with a [latex]76°[/latex] angle of incidence. What is the angle of refraction? The index of refraction from water to glass is 1.1.
Exercise Group
For Problems 47–50, decide which of the following equations are identities. Explain your reasoning.
47.
[latex]\cos x\tan x = \sin x[/latex]
48.
[latex]\sin \theta = 1 - \cos \theta[/latex]
49.
[latex]\tan \phi + \tan \phi = \tan 2\phi[/latex]
50.
[latex]\tan^2 x = \dfrac{\sin^2 x}{1 - \sin^2 x}[/latex]
Exercise Group
For Problems 51–54, use graphs to decide which of the following equations are identities.
51.
[latex]\cos 2\theta = 2 \cos \theta[/latex]
52.
[latex]\cos (x - 90°) = \sin x[/latex]
53.
[latex]\sin 2x = 2\sin x \cos x[/latex]
54.
[latex]\cos (\theta + 90°) = \cos \theta - 1[/latex]
Exercise Group
For Problems 55–58, show that the equation is an identity by transforming the left side into the right side.
55.
[latex]\dfrac{1 - \cos^2 \alpha}{\tan \alpha} = \sin \alpha \cos \alpha[/latex]
56.
[latex]\cos^2 \beta \tan^2 \beta + \cos^2 \beta = 1[/latex]
57.
[latex]\dfrac{\tan \theta - \sin \theta \cos \theta}{\sin \theta \cos \theta} = \sin \theta[/latex]
58.
[latex]\tan \phi - \dfrac{\sin^2 \phi}{\tan \phi} = \tan \phi \sin^2 \phi[/latex]
Exercise Group
For Problems 59–62, simplify, using identities as necessary.
59.
[latex]\tan \theta + \dfrac{\cos \theta}{\sin \theta}[/latex]
60.
[latex]\dfrac{1 - 2\cos^2 \beta}{\sin \beta \cos \beta} + \dfrac{\cos \beta}{\sin \beta}[/latex]
61.
[latex]\dfrac{1}{1 - \sin^2 v} - \tan^2 v[/latex]
62.
[latex]\cos u + (\sin u)(\tan u)[/latex]
Exercise Group
For Problems 63–66, evaluate the expressions without using a calculator.
63.
[latex]\sin 137° - \tan 137° \cdot \cos 137°[/latex]
64.
[latex]\cos^2 8° + \cos 8° \cdot \tan 8° \cdot \sin 8°[/latex]
65.
[latex]\dfrac{1}{\cos^2 54°} - \tan^2 54°[/latex]
66.
[latex]\dfrac{2}{\cos^2 7°} - 2\tan^2 7°[/latex]
Exercise Group
For Problems 67–70, use identities to rewrite each expression.
67.
Write [latex]\tan^2 \beta + 1[/latex] in terms of [latex]\cos^2 \beta{.}[/latex]
68.
Write [latex]2\sin^2 t + \cos t[/latex] in terms of [latex]\cos t{.}[/latex]
69.
Write [latex]\dfrac{\cos x}{\tan x}[/latex] in terms of [latex]\sin x{.}[/latex]
70.
Write [latex]\tan^2 \beta + 1[/latex] in terms of [latex]\cos^2 \beta{.}[/latex]
Exercise Group
For Problems 71–74, find the values of the three trigonometric functions.
71.
[latex]7\tan \beta - 4 = 2, ~~ 180° \lt \beta \lt 270°[/latex]
72.
[latex]3\tan C + 5 = 3, ~~-90° \lt C \lt 0°[/latex]
73.
[latex]5\cos \alpha + 3 = 1, ~~ 90° \lt \alpha \lt 180°[/latex]
74.
[latex]3\sin \theta + 2 = 4, ~~ 90° \lt \beta \lt 180°[/latex]
Exercise Group
For Problems 75–82, solve the equation for [latex]0° \le x \lt 360°{.}[/latex] Round angles to three decimal places if necessary.
75.
[latex]\sin w + 1 = \cos^2 w[/latex]
76.
[latex]\cos^2 \phi - \cos \phi - \sin^2 \phi = 0[/latex]
77.
[latex]\cos x + \sin x = 0[/latex]
78.
[latex]3\sin \theta = \sqrt{3} \cos \theta[/latex]
79.
[latex]2\sin \beta - \tan \beta = 0[/latex]
80.
[latex]6\tan \theta \cos \theta + 6 = 0[/latex]
81.
[latex]\cos^2 t - \sin^2 t = 1[/latex]
82.
[latex]5\cos^2 \beta - 5\sin^2 \beta = -5[/latex]