1.
[latex]250°[/latex]
Chapter 5: Equations and Identities
Exercises: 5.2 Solving Equations
Skills
Practice each skill in the Homework Problems listed:
- Use reference angles
- Solve equations by trial and error
- Use graphs to solve equations
- Solve trigonometric equations for exact values
- Use a calculator to solve trigonometric equations
- Solve trigonometric equations that involve factoring
Suggested Problems
Problems: #4, 8, 12, 16, 26, 30, 46, 38, 50, 56, 64
Exercises Homework 5.2
Exercise Group
For Problems 1–4, find the reference angle. (If you would like to review reference angles, see Section 4.1.)
2.
[latex]145°[/latex]
3.
[latex]320°[/latex]
4.
[latex]-110°[/latex]
Exercise Group
For Problems 5–8, find an angle in each quadrant with the given reference angle.
5.
[latex]18°[/latex]
6.
[latex]35°[/latex]
7.
[latex]52°[/latex]
8.
[latex]78°[/latex]
Exercise Group
For Problems 9–14,
- Evaluate the expression at the given values of the variable.
- Give one solution of the equation.
9.
- [latex]\displaystyle x^3 - 3x^2 + 4;~~~x = -1, 0, 1, 2, 3[/latex]
- [latex]\displaystyle x^3 - 3x^2 + 4 = 0[/latex]
10.
- [latex]\displaystyle \\\sqrt{x} + \\\sqrt{2x+1};~~~x = 0, 2, 4, 6[/latex]
- [latex]\displaystyle \\\sqrt{x} + \\\sqrt{2x+1} = 5[/latex]
11.
- [latex]\displaystyle \sin \theta + \cos \theta;~~~\theta = 0°, 30°, 45°, 60°[/latex]
- [latex]\displaystyle \sin \theta + \cos \theta = \\\sqrt{2}[/latex]
12.
- [latex]\displaystyle \sin^2 \alpha - \cos \alpha;~~~\alpha = 45°, 90°, 135°, 180°[/latex]
- [latex]\displaystyle \sin^2 \alpha - \cos \alpha = 1[/latex]
13.
- [latex]\displaystyle \sin \beta + 2\cos^2 \beta;~~~\beta = 210°, 225°, 240°, 270°[/latex]
- [latex]\displaystyle \sin \beta + 2\cos^2 \beta = -1[/latex]
14.
- [latex]\displaystyle 3\cos^2 \phi - \sin^2 \phi;~~~\phi = 270°, 300°, 315°, 330°[/latex]
- [latex]\displaystyle 3\cos^2 \phi - \sin^2 \phi = 2[/latex]
Exercise Group
For Problems 15–18, use a graph to solve the equation. Check your solution by substitution.
15.
[latex]\dfrac{-1}{3}x^2 + \dfrac{2}{3}x + 5 = 0[/latex]
16.
[latex]0.0625x^2 + 0.5 x = -1[/latex]
17.
[latex]x^3 + 2x^2 - 6 = 2x^2 + 7x[/latex]
18.
[latex]8 - 12x + 6x^2 - x^3[/latex]
Exercise Group
For Problems 19–32, solve the equation exactly for [latex]0° \le\theta\lt 360°{.}[/latex]
19.
[latex]3\tan \theta = \\\sqrt{3}[/latex]
20.
[latex]7\sin \theta + 11 = 11[/latex]
21.
[latex]3 = 5 - 4\cos \theta[/latex]
22.
[latex]6\tan \theta + 21 = 15[/latex]
23.
[latex]8\sin \theta + 5 = 1[/latex]
24.
[latex]9\cos \theta + 15 = 6[/latex]
25.
[latex]0 = \\\sqrt{2} + 2\sin \theta[/latex]
26.
[latex]\\\sqrt{3}\cos \theta = -\dfrac{3}{2}[/latex]
27.
[latex]\cos^2 \theta - 1 = 0[/latex]
28.
[latex]1 - \sin^2 \theta = 0[/latex]
29.
[latex]4\sin^2 \theta - 3 = 0[/latex]
30.
[latex]0 = 1 - 2\cos^2 \theta[/latex]
31.
[latex]1 - \tan^2 \theta = 0[/latex]
32.
[latex]0 = 6 \tan^2 \theta - 2[/latex]
Exercise Group
For Problems 33–38, solve the equation for [latex]0° \le\theta\lt 360°{.}[/latex] Round your answers to two decimal places.
33.
[latex]\dfrac{1}{2}\tan \theta - 1 = -3[/latex]
34.
[latex]3\tan \theta - 2 = 4[/latex]
35.
[latex]3 = 5\cos \theta[/latex]
36.
[latex]4 = 6\sin \theta[/latex]
37.
[latex]7 \sin \theta + 2 = 1[/latex]
38.
[latex]2 = 5 - \dfrac{1}{3} \tan \theta[/latex]
Exercise Group
For Problems 39–46, use a graph to estimate the solutions for angles between [latex]0°[/latex] and [latex]360°{.}[/latex] Solve the equation algebraically.
39.
[latex]7 - \tan A = 8[/latex]
40.
[latex]6 = 8\tan w - 2[/latex]
41.
[latex]5 = 1 - 8\sin \phi[/latex]
42.
[latex]9 - 4\sin t = 13[/latex]
43.
[latex]2\cos B - 2 = -2[/latex]
44.
[latex]2 - 6\cos u = 5[/latex]
45.
[latex]3 = 2\sin \theta + 4[/latex]
46.
[latex]5 = 3\cos x + 5[/latex]
Exercise Group
For Problems 47–52, use a graph to estimate the solutions for angles between [latex]0°[/latex] and [latex]360°{.}[/latex] Solve the equation algebraically, rounding angles to the nearest degree.
47.
[latex]8\sin t + 7 = 4[/latex]
48.
[latex]9 - 6\cos A = 5[/latex]
49.
[latex]5\tan B - 4 = -2[/latex]
50.
[latex]3 - 10\tan C = -11[/latex]
51.
[latex]1 + 6\cos \phi = -4[/latex]
52.
[latex]4\sin u - 2 = -1[/latex]
Exercise Group
For Problems 53–64, solve the equation for [latex]0° \le\theta\le 360°{.}[/latex] Round angles to two decimal places.
53.
[latex]6\cos^2 \theta = 2[/latex]
54.
[latex]2 - 7\sin^2 \phi = 1[/latex]
55.
[latex]5\sin^2 \theta + \sin \theta = 0[/latex]
56.
[latex]4\tan^2 \theta = \tan \theta[/latex]
57.
[latex]2\cos^2 \theta + \cos \theta - 1 = 0[/latex]
58.
[latex]\tan^2 \theta - 5\tan \theta + 6 = 0[/latex]
59.
[latex]6\tan^2 \theta - \tan \theta - 1 = 0[/latex]
60.
[latex]10\cos^2 \theta - 7 \cos \theta + 1 = 0[/latex]
61.
[latex]\tan^2 \theta - 2\tan \theta = 15[/latex]
62.
[latex]\tan \theta = \tan^2 \theta - 0[/latex]
63.
[latex]\cos^2 \theta - 4\cos \theta + 3 = 0[/latex]
64.
[latex]\sin^2 \theta + 8\sin \theta + 7 = 0[/latex]
Exercise Group
For Problems 65–68, use Snell’s Law to answer the question.
65.
A light ray passes from water to glass, with a [latex]19°[/latex] angle of incidence. What is the angle of refraction?
66.
A light ray passes from water to glass, with an [latex]82°[/latex] angle of incidence. What is the angle of refraction?
67.
A light ray passes from water to glass, with a [latex]32°[/latex] angle of refraction. What is the angle of incidence?
68.
A light ray passes from water to glass, with a [latex]58°[/latex] angle of refraction. What is the angle of incidence?
69.
- Use your calculator to graph the function [latex]y = \tan \theta[/latex] in the ZTrig window (press [latex]\boxed{{ZOOM}}~ \boxed{{7}}[/latex]), along with the horizontal line [latex]y = 2{.}[/latex] Use the intersect feature to verify that the solutions of the equation [latex]\tan \theta = 2[/latex] differ by [latex]180°{.}[/latex]
- Repeat part (a) with the horizontal line [latex]y = -2[/latex] to verify that the solutions of the equation [latex]\tan \theta = -2[/latex] differ by [latex]180°{.}[/latex]
70.
- What is the angle in the third quadrant with reference angle [latex]\theta[/latex]? Show this angle differs by [latex]180°{.}[/latex] Explain how this fact shows that the solutions of [latex]\tan \theta = k{,}[/latex] for [latex]k \gt 0{,}[/latex] differ by [latex]180°{.}[/latex]
- What is the angle in the second quadrant with reference angle [latex]\theta[/latex]? What is the angle in the fourth quadrant with reference angle [latex]\theta[/latex]? Show that these two angles differ by [latex]180°{.}[/latex] Explain how this fact shows that the solutions [latex]\tan \theta = k{,}[/latex] for [latex]k \lt 0{,}[/latex] differ by [latex]180°{.}[/latex]
