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Chapter 4: Trig Functions

Exercises: 4.2 Graphs of Trigonometric Functions

Skills

Practice each skill in the Homework Problems listed:

  • Find coordinates
  • Use bearings to determine position
  • Sketch graphs of the sine and cosine functions
  • Find the coordinates of points on a sine or cosine graph
  • Use function notation
  • Find reference angles
  • Solve equations graphically
  • Graph the tangent function
  • Find and use the angle of inclination of a line

 

Suggested Problems

Problems: #4, 12, 18, 26, 32, 30, 40, 48, 50, 56, 60, 62

Homework 4.2

Exercise Group

For Problems 1–6, find exact values for the coordinates of the point.

1.

315 angle in standard position, r=8

2.

225 angle in standard position, r=12

3.

240 angle in standard position, r=20

4.

330 angle in standard position, r=4

5.

150 angle in standard position, r=15

6.

120 angle in standard position, r=9

Exercise Group

For Problems 7–12, find the coordinates of the point, rounded to hundredths.

7.

258 angle in standard position, r=6

8.

312 angle in standard position, r=16

9.

296 angle in standard position, r=13

10.

341 angle in standard position, r=10

11.

204 angle in standard position, r=20

12.

106 angle in standard position, r=7

Exercise Group

For Problems 13–18, a ship sails from the seaport on the given bearing for the given distance.

  1. Make a sketch showing the ship’s current location relative to the seaport.
  2. How far east or west of the seaport is the ship’s present location? How far north or south?
13.

[latex]36°{,}[/latex] 26 miles

14.

[latex]124°{,}[/latex] 80 km

15.

[latex]230°{,}[/latex] 120 km

16.

[latex]318°{,}[/latex] 75 miles

17.

[latex]285°{,}[/latex] 32 km

18.

[latex]192°{,}[/latex] 260 miles

19.

Estimate the [latex]x[/latex]-coordinate of the points on the unit circle designated by each angle, and complete the table. (Hint: Use symmetry in the second, third, and fourth quadrants.)

unit circle

Angle [latex]0°[/latex] [latex]10°[/latex] [latex]20°[/latex] [latex]30°[/latex] [latex]40°[/latex] [latex]50°[/latex] [latex]60°[/latex] [latex]70°[/latex] [latex]80°[/latex] [latex]90°[/latex]
[latex]x[/latex]-coordinate [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] [latex]\hphantom{0000}[/latex]
Angle [latex]100°[/latex] [latex]110°[/latex] [latex]120°[/latex] [latex]130°[/latex] [latex]140°[/latex] [latex]150°[/latex] [latex]160°[/latex] [latex]170°[/latex] [latex]180°[/latex]
[latex]x[/latex]-coordinate [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]
Angle [latex]190°[/latex] [latex]200°[/latex] [latex]210°[/latex] [latex]220°[/latex] [latex]230°[/latex] [latex]240°[/latex] [latex]250°[/latex] [latex]260°[/latex] [latex]270°[/latex]
[latex]x[/latex]-coordinate [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]
Angle [latex]280°[/latex] [latex]290°[/latex] [latex]300°[/latex] [latex]310°[/latex] [latex]320°[/latex] [latex]330°[/latex] [latex]340°[/latex] [latex]350°[/latex] [latex]360°[/latex]
[latex]x[/latex]-coordinate [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]
20.

Estimate the [latex]y[/latex]-coordinate of the points on the unit circle designated by each angle, and complete the table. (Hint: Use symmetry in the second, third, and fourth quadrants.) unit circle

Angle [latex]0°[/latex] [latex]10°[/latex] [latex]20°[/latex] [latex]30°[/latex] [latex]40°[/latex] [latex]50°[/latex] [latex]60°[/latex] [latex]70°[/latex] [latex]80°[/latex] [latex]90°[/latex]
[latex]x[/latex]-coordinate [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] [latex]\hphantom{0000}[/latex]
Angle [latex]100°[/latex] [latex]110°[/latex] [latex]120°[/latex] [latex]130°[/latex] [latex]140°[/latex] [latex]150°[/latex] [latex]160°[/latex] [latex]170°[/latex] [latex]180°[/latex]
[latex]x[/latex]-coordinate [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]
Angle [latex]190°[/latex] [latex]200°[/latex] [latex]210°[/latex] [latex]220°[/latex] [latex]230°[/latex] [latex]240°[/latex] [latex]250°[/latex] [latex]260°[/latex] [latex]270°[/latex]
[latex]x[/latex]-coordinate [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]
Angle [latex]280°[/latex] [latex]290°[/latex] [latex]300°[/latex] [latex]310°[/latex] [latex]320°[/latex] [latex]330°[/latex] [latex]340°[/latex] [latex]350°[/latex] [latex]360°[/latex]
[latex]x[/latex]-coordinate [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]
21.
  1. Draw vertical line segments from the unit circle to the [latex]x[/latex]-axis to illustrate the [latex]y[/latex]-coordinate of each point designated by the angles, [latex]0°[/latex] to [latex]90°{,}[/latex] shown on the figure below.
  2. Transfer your vertical line segments to the appropriate position on the grid below.
  3. Repeat parts (a) and (b) for the other three quadrants.
  4. Connect the tops of the segments to sketch a graph of [latex]y = \sin \theta{.}[/latex]

unit circle and blank graph

22.
  1. Draw horizontal line segments from the unit circle to the [latex]y[/latex]-axis to illustrate the [latex]x[/latex]-coordinate of each point designated by the angles, [latex]0°[/latex] to [latex]90°{,}[/latex] shown on the figure below.
  2. Transfer your horizontal line segments into vertical line segments at the appropriate position on the grid below.
  3. Repeat parts (a) and (b) for the other three quadrants.
  4. Connect the tops of the segments to sketch a graph of [latex]y = \cos \theta{.}[/latex]

unit circle and blank graph

23.
  1. Prepare a graph with the horizontal axis scaled from [latex]0°[/latex] to [latex]360°[/latex] in multiples of [latex]45°{.}[/latex]
  2. Sketch a graph of [latex]f(\theta) = \sin \theta[/latex] by plotting points for multiples of [latex]45°{.}[/latex]
24.
  1. Prepare a graph with the horizontal axis scaled from [latex]0°[/latex] to [latex]360°[/latex] in multiples of [latex]45°{.}[/latex]
  2. Sketch a graph of [latex]f(\theta) = \cos \theta[/latex] by plotting points for multiples of [latex]45°{.}[/latex]
25.
  1. Prepare a graph with the horizontal axis scaled from [latex]0°[/latex] to [latex]360°[/latex] in multiples of [latex]30°{.}[/latex]
  2. Sketch a graph of [latex]f(\theta) = \cos \theta[/latex] by plotting points for multiples of [latex]30°{.}[/latex]
26.
  1. Prepare a graph with the horizontal axis scaled from [latex]0°[/latex] to [latex]360°[/latex] in multiples of [latex]30°{.}[/latex]
  2. Sketch a graph of [latex]f(\theta) = \sin \theta[/latex] by plotting points for multiples of [latex]30°{.}[/latex]

Exercise Group

For Problems 27–30, give the coordinates of each point on the graph of [latex]f(\theta) = \sin \theta[/latex] or [latex]f(\theta) = \cos \theta.[/latex]

27.

sine graph

28.

sine graph

29.

cosine graph

30.

cosine graph

31.

Make a short table of values like the one shown and sketch the function by hand. Be sure to label the [latex]x[/latex]-axis and [latex]y[/latex]-axis appropriately.

[latex]\theta[/latex] [latex]0°[/latex] [latex]90°[/latex] [latex]180°[/latex] [latex]270°[/latex] [latex]360°[/latex]
[latex]f(\theta)[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
  1. [latex]\displaystyle f(\theta) = \sin \theta[/latex]
  2. [latex]\displaystyle f(\theta) = \cos \theta[/latex]
32.

One of these graphs is [latex]y = A \sin k\theta{,}[/latex] and the other is [latex]y = A \cos k\theta{.}[/latex] Explain how you know which is which. sinusoidal graphs

Exercise Group

For Problems 33–40, evaluate the expression for [latex]f(\theta) = \sin \theta[/latex] and [latex]g(\theta) = \cos \theta{.}[/latex]

33.

[latex]3 + f(30°)[/latex]

34.

[latex]3 f(30°)[/latex]

35.

[latex]4g(225°) - 1[/latex]

36.

[latex]-4 + 2g(225°) - 1[/latex]

37.

[latex]-2f(3\theta){,}[/latex] for [latex]\theta = 90°[/latex]

38.

[latex]6f(\dfrac{\theta}{2}){,}[/latex] for [latex]\theta = 90°[/latex]

39.

[latex]8 - 5g(\dfrac{\theta}{3}){,}[/latex] for [latex]\theta = 360°[/latex]

40.

[latex]1 - 4g(4\theta){,}[/latex] for [latex]\theta = 135°[/latex]

41.

The graph shows your height as a function of angle as you ride the Ferris wheel. For each location A-E on the Ferris wheel, mark the corresponding point on the graph. circle and graph

42.

The graph shows your height as a function of angle as you ride the Ferris wheel. For each location F-J on the graph, mark the corresponding point on the Ferris wheel. circle and graph

43.

The graph shows the horizontal displacement of your foot from the center of the chain gear as you pedal a bicycle. For each location K-O on the chain gear, mark the corresponding point on the graph.
circle and graph

44.

The graph shows the horizontal displacement of your foot from the center of the chain gear as you pedal a bicycle. For each location P-T on the graph, mark the corresponding point on the chain gear.
circle and graph

Exercise Group

For Problems 45–46, use the grid provided below. unit circle on grid

45.

Draw two different angles [latex]\alpha[/latex] and [latex]\beta[/latex] in standard position whose sine is [latex]0.6{.}[/latex]

  1. Use a protractor to measure [latex]\alpha[/latex] and [latex]\beta{.}[/latex]
  2. Find the reference angles for both [latex]\alpha[/latex] and [latex]\beta{.}[/latex] Draw in the reference triangles.
46.

Draw two different angles [latex]\theta[/latex] and [latex]\phi[/latex] in standard position whose sine is [latex]-0.8{.}[/latex]

  1. Use a protractor to measure [latex]\theta[/latex] and [latex]\phi{.}[/latex]
  2. Find the reference angles for both [latex]\theta[/latex] and [latex]\phi{.}[/latex] Draw in the reference triangles.

Exercise Group

For Problems 47–48, use the grid provided below. unit circle on grid

47.

Draw two different angles [latex]\alpha[/latex] and [latex]\beta[/latex] in standard position whose cosine is [latex]0.3{.}[/latex]

  1. Use a protractor to measure [latex]\alpha[/latex] and [latex]\beta{.}[/latex]
  2. Find the reference angles for both [latex]\alpha[/latex] and [latex]\beta{.}[/latex] Draw in the reference triangles.
48.

Draw two different angles [latex]\theta[/latex] and [latex]\phi[/latex] in standard position whose cosine is [latex]-0.4{.}[/latex]

  1. Use a protractor to measure [latex]\theta[/latex] and [latex]\phi{.}[/latex]
  2. Find the reference angles for both [latex]\theta[/latex] and [latex]\phi{.}[/latex] Draw in the reference triangles.

Exercise Group

For Problems 49–56, use the graphs to estimate the solutions to the equations. Show your work on the graph. sine graph cosine graph

49.

[latex]\sin \theta = 0.6[/latex]

50.

[latex]\sin \theta = -0.8[/latex]

51.

[latex]\cos \theta = 0.3[/latex]

52.

[latex]\cos \theta = -0.4[/latex]

53.

[latex]\sin \theta = -0.2[/latex]

54.

[latex]\sin \theta = 1.2[/latex]

55.

[latex]\cos \theta = -0.9[/latex]

56.

[latex]\cos \theta = -1.1[/latex]

57.
  1. Fill in the table for values of [latex]\tan \theta{.}[/latex] Round your answers to three decimal places.
    [latex]\theta[/latex] [latex]81°[/latex] [latex]82°[/latex] [latex]83°[/latex] [latex]84°[/latex] [latex]85°[/latex] [latex]86°[/latex] [latex]87°[/latex] [latex]88°[/latex] [latex]89°[/latex]
    [latex]\tan \theta[/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]
  2. What happens to [latex]\tan \theta[/latex] as [latex]\theta[/latex] increases toward [latex]90°{?}[/latex]
  3. Fill in the table for values of [latex]\tan \theta{.}[/latex] Round your answers to three decimal places.
    [latex]\theta[/latex] [latex]99°[/latex] [latex]98°[/latex] [latex]97°[/latex] [latex]96°[/latex] [latex]95°[/latex] [latex]94°[/latex] [latex]93°[/latex] [latex]92°[/latex] [latex]91°[/latex]
    [latex]\tan \theta[/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]
  4. What happens to [latex]\tan \theta[/latex] as [latex]\theta[/latex] decreases toward [latex]90°{?}[/latex]
  5. What value does your calculator give for [latex]\tan 90°{?}[/latex] Why?
58.
  1. Fill in the table with exact values of [latex]\tan \theta{.}[/latex] Then give decimal approximations to two places.
    [latex]\theta[/latex] [latex]0°[/latex] [latex]30°[/latex] [latex]45°[/latex] [latex]60°[/latex] [latex]90°[/latex] [latex]120°[/latex] [latex]135°[/latex] [latex]150°[/latex] [latex]180°[/latex]
    [latex]\tan \theta[/latex] (exact) [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]
    [latex]\tan \theta[/latex] (approx.) [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]
  2. Fill in the table with exact values of [latex]\tan \theta{.}[/latex] Then give decimal approximations to two places.
    [latex]\theta[/latex] [latex]180°[/latex] [latex]210°[/latex] [latex]225°[/latex] [latex]240°[/latex] [latex]270°[/latex] [latex]300°[/latex] [latex]315°[/latex] [latex]330°[/latex] [latex]360°[/latex]
    [latex]\tan \theta[/latex] (exact) [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]
    [latex]\tan \theta[/latex] (approx.) [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]
  3. Plot the points from the tables and sketch a graph of [latex]f(\theta) = \tan \theta{.}[/latex]tan grid
59.
  • Sketch by hand a graph of [latex]y = \tan \theta[/latex] for [latex]-180° \le \theta \le 180°{.}[/latex]
60.
  • Use your calculator to graph [latex]y = \tan \theta[/latex] in the ZTrig window (press ZOOM 7). Sketch the result. On your sketch, mark scales on the axes and include dotted lines for the vertical asymptotes.

Exercise Group

For Problems 61–64, find the angle of inclination of the line.

61.

[latex]y = \dfrac{5}{4}x - 3[/latex]

62.

[latex]y = 6 + \dfrac{2}{9}x[/latex]

63.

[latex]y = -2 - \dfrac{3}{8}x[/latex]

64.

[latex]y = \dfrac{-7}{2}x + 1[/latex]

Exercise Group

For Problems 65–68, find an equation for the line passing through the given point with angle of inclination [latex]\alpha{.}[/latex]

65.

[latex](3,-5), ~\alpha = 28°[/latex]

66.

[latex](-2,6), ~\alpha = 67°[/latex]

67.

[latex](-8,12), ~\alpha = 112°[/latex]

68.

[latex](-4,-1), ~\alpha = 154°[/latex]

69.

The slope of a line is a function of its angle of inclination, [latex]m = f(\alpha){.}[/latex] Complete the table and sketch a graph of the function.

[latex]\alpha[/latex] [latex]0°[/latex] [latex]15°[/latex] [latex]30°[/latex] [latex]45°[/latex] [latex]60°[/latex] [latex]75°[/latex] [latex]90°[/latex] [latex]105°[/latex] [latex]120°[/latex] [latex]135°[/latex] [latex]150°[/latex] [latex]165°[/latex] [latex]180°[/latex]
[latex]m[/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] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
  1. What happens to the slope of the line as [latex]\alpha[/latex] increases toward [latex]90°{?}[/latex]
  2. What happens to the slope of the line as [latex]\alpha[/latex] decreases toward [latex]90°{?}[/latex]
70.

The angle of inclination of a line is a function of its slope, [latex]\alpha = g(m){.}[/latex] Complete the table and sketch a graph of the function.

[latex]m[/latex] [latex]-20[/latex] [latex]-10[/latex] [latex]-5[/latex] [latex]-2[/latex] [latex]-1[/latex] [latex]-0.75[/latex] [latex]-0.5[/latex] [latex]-0.25[/latex] [latex]0[/latex]
[latex]\alpha[/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]
[latex]m[/latex] [latex]0.25[/latex] [latex]0.5[/latex] [latex]0.75[/latex] [latex]1[/latex] [latex]2[/latex] [latex]5[/latex] [latex]10[/latex] [latex]20[/latex]
[latex]\alpha[/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]
  1. What happens to the angle of inclination as the slope increases toward infinity?
  2. What happens to the angle of inclination as the slope decreases toward negative infinity?

 

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