1.
[latex]\left(\sqrt{a} + \sqrt{b}\right)^2 = a + b[/latex]
Chapter 5: Equations and Identities
Exercises: 5.3 Trigonometric Identities
Skills
Practice each skill in the Homework Problems listed:
- Recognize identities
- Verify identities
- Rewrite expressions using identities
- Use identities to evaluate expressions
- Solve trigonometric equations
- Given one trig ratio, find the others
Suggested Problems
Problems: #4, 8, 16, 44, 22, 24, 74, 28, 32, 50, 36, 58, 60, 70
Exercises Homework 5.3
Exercise Group
For Problems 1–8, decide which of the following equations are identities. Explain your reasoning.
2.
[latex]\sqrt{a^2 - b^2} = a - b[/latex]
3.
[latex]\dfrac{1}{a + b} = \dfrac{1}{a} + \dfrac{1}{b}[/latex]
4.
[latex]\dfrac{a + b}{a} = b[/latex]
5.
[latex]\tan (\alpha + \beta) = \dfrac{\sin (\alpha + \beta)}{\cos (\alpha + \beta)}[/latex]
6.
[latex]\dfrac{1}{\tan \theta} = \dfrac{\cos \theta}{\sin \theta}[/latex]
7.
[latex](1 + \tan \theta)^2 = 1 + \tan^2 \theta[/latex]
8.
[latex]\sqrt{1 - \sin^2 \phi} = 1 - \sin \phi[/latex]
Exercise Group
For Problems 9–16, use graphs to decide which of the following equations are identities.
9.
[latex]\sin 2t = 2 \sin t[/latex]
10.
[latex]\cos \theta + \sin \theta = 1[/latex]
11.
[latex]\sin (30° + \beta) = \dfrac{1}{2} + \sin \beta[/latex]
12.
[latex]\cos (90° - C) = \sin C[/latex]
13.
[latex]\tan (90° - \theta) = \dfrac{1}{\tan \theta}[/latex]
14.
[latex]\tan 2\theta = \dfrac{2\tan \theta}{1 - \tan^2 \theta}[/latex]
15.
[latex]\dfrac{\tan^2 x}{1 + \tan^2 x} = \sin^2 x[/latex]
16.
[latex]\tan x + \dfrac{1}{\tan x} = \sin x \cos x[/latex]
Exercise Group
For Problems 17–26, show that the equation is an identity by transforming the left side into the right side.
17.
[latex](1 + \sin w)(1 - \sin w) = \cos^2 w[/latex]
18.
[latex](\cos \theta - 1)(\cos \theta + 1) = -\sin^2 \theta[/latex]
19.
[latex](\cos \theta - \sin \theta)^2 = 1 - 2 \sin \theta \cos \theta[/latex]
20.
[latex]\sin^2 x - \cos^2 x = 1 - 2\cos^2 x[/latex]
21.
[latex]\tan \theta \cos \theta = \sin \theta[/latex]
22.
[latex]\dfrac{\sin \mu}{\tan \mu} = \cos \mu[/latex]
23.
[latex]\cos^4 x - \sin^4 x = \cos^2 x - \sin^2 x[/latex]
24.
[latex]1 - 2\cos^2 v + \cos^4 v = \sin^4 v[/latex]
25.
[latex]\dfrac{\sin u}{1 + \cos u} = \dfrac{1 - \cos u}{\sin u}[/latex]
Multiply numerator and denominator of the left side by [latex]1 - \cos u{.}[/latex]
26.
[latex]\dfrac{\sin v}{1 - \cos v} = \dfrac{\tan v(1 + \sin v)}{\cos v}[/latex]
Multiply numerator and denominator of the left side by [latex]1 + \sin v{.}[/latex]
Exercise Group
For Problems 27–34, simplify, using identities as necessary.
27.
[latex]\dfrac{1}{\cos^2 \beta}- \dfrac{\sin^2 \beta}{\cos^2 \beta}[/latex]
28.
[latex]\dfrac{1}{\sin^2 \phi}- \dfrac{1}{\tan^2 \phi}[/latex]
29.
[latex]\cos^2 \alpha (1 + \tan^2 \alpha)[/latex]
30.
[latex]\cos^3 \phi + \sin^2 \phi \cos \phi[/latex]
31.
[latex]\tan^2 A - \tan^2 A \sin^2 A[/latex]
32.
[latex]\cos^2 B \tan^2 B + \cos^2 B[/latex]
33.
[latex]\dfrac{1 - \cos^2 z}{\cos^2 z}[/latex]
34.
[latex]\dfrac{\sin t}{\cos t \tan t}[/latex]
Exercise Group
For Problems 35–40, evaluate without using a calculator.
35.
[latex]3\cos^2 1.7° + 3\sin^2 1.7°[/latex]
36.
[latex]4 - \cos^2 338° - \sin^2 338°[/latex]
37.
[latex](\cos^2 20° + \sin^2 20°)^4[/latex]
38.
[latex]\dfrac{18}{\cos^2 17° + \sin^2 17°}[/latex]
39.
[latex]\dfrac{6}{\cos^2 53°} - 6 \tan^2 53°[/latex]
40.
[latex]\dfrac{1}{\sin^2 102°} - \dfrac{\cos^2 102°}{\sin^2 102°}[/latex]
Exercise Group
For Problems 41–46, one side of an identity is given. Graph the expression and make a conjecture about the other side of the identity.
41.
[latex]2\cos^2 \theta - 1 = ?[/latex]
42.
[latex]1 - 2\sin^2 \left(\dfrac{\theta}{2}\right) = ?[/latex]
43.
[latex]1 - \dfrac{\sin^2 x}{1 + \cos x} = ?[/latex]
44.
[latex]\dfrac{\sin x}{\sqrt{1 - \sin^2 x}} = ?[/latex]
45.
[latex]2\tan t \cos^2 t = ?[/latex]
46.
[latex]\dfrac{2 \tan t}{1 - \tan^2 t} = ?[/latex]
Exercise Group
For Problems 47–50, use identities to rewrite each expression.
47.
[latex]2 - \cos^2 \theta + 2 \sin \theta~~~[/latex] as an expression in [latex]\sin \theta[/latex] only
48.
[latex]3\sin^2 B + 2\cos B - 4~~~[/latex] as an expression in [latex]\cos B[/latex] only
49.
[latex]\cos^2 \phi - 2\sin^2 \phi~~~[/latex] as an expression in [latex]\cos \phi[/latex] only
50.
[latex]\cos^2 \phi \sin^2 \phi~~~[/latex] as an expression in [latex]\sin \phi[/latex] only
Exercise Group
For Problems 51–58, solve the equation for [latex]0° \le\theta\le 360°{.}[/latex] Round angles to three decimal places if necessary.
51.
[latex]\cos \theta - \sin^2 \theta + 1 = 0[/latex]
52.
[latex]4\sin \theta + 2\cos^2 \theta - 3 = -1[/latex]
53.
[latex]1 - \sin \theta - 2\cos^2 \theta = 0[/latex]
54.
[latex]3\cos^2 \theta - \sin^2 \theta = 2[/latex]
55.
[latex]2\cos \theta \tan \theta + 1 = 0[/latex]
56.
[latex]\cos \theta - \sin \theta = 0[/latex]
57.
[latex]\dfrac{1}{3}\cos \theta = \sin \theta[/latex]
58.
[latex]5\sin C = 2\cos C[/latex]
Exercise Group
For Problems 59–62, use identities to find exact values for the other two trig ratios.
59.
[latex]\cos A = \dfrac{12}{13}~~~[/latex] and [latex]~270° \lt\ A \lt 360°[/latex]
60.
[latex]\sin B = \dfrac{-3}{5}~~~[/latex] and [latex]~180° \lt\ B \lt 270°[/latex]
61.
[latex]\sin \phi = \dfrac{1}{7}~~~[/latex] and [latex]~90° \lt\ \phi \lt 180°[/latex]
62.
[latex]\cos t = \dfrac{-2}{3}~~~[/latex] and [latex]~180° \lt\ t \lt 270°[/latex]
Exercise Group
For Problems 63–66, use the identity below to find the sine and cosine of the angle.
[latex]{1 + \tan^2 \theta = \dfrac{1}{\cos^2 \theta}}[/latex]
63.
[latex]\tan \theta = -\dfrac{1}{2}~~~[/latex] and [latex]~270° \lt\ \theta \lt 360°[/latex]
64.
[latex]\tan \theta = 2~~~[/latex] and [latex]~180° \lt\ \theta \lt 270°[/latex]
65.
[latex]\tan \theta = \dfrac{3}{4}~~~[/latex] and [latex]~180° \lt\ \theta \lt 270°[/latex]
66.
[latex]\tan \theta = -3~~~[/latex] and [latex]~90° \lt\ \theta \lt 180°[/latex]
Exercise Group
For Problems 67–72, find exact values for the sine, cosine, and tangent of the angle.
67.
[latex]2\cos A + 9 = 8~~~[/latex] and [latex]~90° \lt\ A \lt 180°[/latex]
68.
[latex]25\sin B + 8 = -12~~~[/latex] and [latex]~180° \lt\ B \lt 270°[/latex]
69.
[latex]8\tan \beta + 5 = -11~~~[/latex] and [latex]~90° \lt\ \beta \lt 180°[/latex]
70.
[latex]6(\tan \beta - 4) = -24~~~[/latex] and [latex]~90° \lt\ \beta \lt 270°[/latex]
71.
[latex]\tan^2 C - \dfrac{1}{4} = 0~~~[/latex] and [latex]~0° \lt\ C \lt 180°[/latex]
72.
[latex]4\cos^2 A - \cos A = 0~~~[/latex] and [latex]~00° \lt\ A \lt 180°[/latex]
Exercise Group
For Problems 73–76, prove the identity by rewriting tangents in terms of sines and cosines. (These problems involve simplifying complex fractions. See the Algebra Refresher to review this skill.)
73.
[latex]\dfrac{\tan \alpha}{1 + \tan \alpha} = \dfrac{\sin \alpha}{\sin \alpha + \cos \alpha}[/latex]
74.
[latex]\dfrac{1 - \tan u}{1 + \tan u} = \dfrac{\cos u - \sin u}{\cos u + \sin u}[/latex]
75.
[latex]\dfrac{1 + \tan^2 \beta}{1 - \tan^2 \beta} = \dfrac{1}{\cos^2 \beta - \sin^2 \beta}[/latex]
76.
[latex]\tan^2 v - \sin^2 v = \tan^2 v \sin^2 v[/latex]
77.
Prove the Pythagorean identity [latex]\cos^2 \theta + \sin^2 \theta = 1[/latex] by carrying out the following steps. Sketch an angle [latex]\theta[/latex] in standard position and label a point [latex](x,y)[/latex] on the terminal side, at a distance [latex]r[/latex] from the vertex.
- Begin with the equation [latex]\sqrt{x^2 + y^2} = r{,}[/latex] and square both sides.
- Divide both sides of your equation from part (a) by [latex]r^2{.}[/latex]
- Write the left side of the equation as the sum of the squares of two fractions.
- Substitute the appropriate trigonometric ratio for each fraction.
78.
Prove the tangent identity [latex]\tan \theta = \dfrac{\sin \theta}{\cos \theta}[/latex] by carrying out the following steps. Sketch an angle [latex]\theta[/latex] in standard position and label a point [latex](x,y)[/latex] on the terminal side, at a distance [latex]r[/latex] from the vertex.
- Write [latex]\sin \theta[/latex] in terms of [latex]y[/latex] and [latex]r{,}[/latex] and solve for [latex]y{.}[/latex]
- Write [latex]\cos \theta[/latex] in terms of [latex]x[/latex] and [latex]r{,}[/latex] and solve for [latex]x{.}[/latex]
- Write [latex]\tan \theta[/latex] in terms of [latex]x[/latex] and [latex]y{,}[/latex] then substitute your results from parts (a) and (b).
- Simplify your fraction in part (c).