Exercises: 5.2 Solving Equations

Bimal Kunwor; Donna Densmore; Jared Eusea; Yi Zhen

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

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.)

1.

[latex]250°[/latex]

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]