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
Homework 4.2
Exercise Group
For Problems 1–6, find exact values for the coordinates of the point.
2.
3.
4.
5.
6.
Exercise Group
For Problems 7–12, find the coordinates of the point, rounded to hundredths.
7.
8.
9.
10.
11.
12.
Exercise Group
For Problems 13–18, a ship sails from the seaport on the given bearing for the given distance.
- Make a sketch showing the ship’s current location relative to the seaport.
- 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.)
| 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.)
| 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.
- 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.
- Transfer your vertical line segments to the appropriate position on the grid below.
- Repeat parts (a) and (b) for the other three quadrants.
- Connect the tops of the segments to sketch a graph of [latex]y = \sin \theta{.}[/latex]
22.
- 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.
- Transfer your horizontal line segments into vertical line segments at the appropriate position on the grid below.
- Repeat parts (a) and (b) for the other three quadrants.
- Connect the tops of the segments to sketch a graph of [latex]y = \cos \theta{.}[/latex]
23.
- Prepare a graph with the horizontal axis scaled from [latex]0°[/latex] to [latex]360°[/latex] in multiples of [latex]45°{.}[/latex]
- Sketch a graph of [latex]f(\theta) = \sin \theta[/latex] by plotting points for multiples of [latex]45°{.}[/latex]
24.
- Prepare a graph with the horizontal axis scaled from [latex]0°[/latex] to [latex]360°[/latex] in multiples of [latex]45°{.}[/latex]
- Sketch a graph of [latex]f(\theta) = \cos \theta[/latex] by plotting points for multiples of [latex]45°{.}[/latex]
25.
- Prepare a graph with the horizontal axis scaled from [latex]0°[/latex] to [latex]360°[/latex] in multiples of [latex]30°{.}[/latex]
- Sketch a graph of [latex]f(\theta) = \cos \theta[/latex] by plotting points for multiples of [latex]30°{.}[/latex]
26.
- Prepare a graph with the horizontal axis scaled from [latex]0°[/latex] to [latex]360°[/latex] in multiples of [latex]30°{.}[/latex]
- 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 -27.[/latex]
27.
28.
29.
30.
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] |
- [latex]\displaystyle f(\theta) = \sin \theta[/latex]
- [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.
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.
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.
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.
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.
Exercise Group
For Problems 45–46, use the grid provided below.
45.
Draw two different angles [latex]\alpha[/latex] and [latex]\beta[/latex] in standard position whose sine is [latex]0.6{.}[/latex]
- Use a protractor to measure [latex]\alpha[/latex] and [latex]\beta{.}[/latex]
- 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]
- Use a protractor to measure [latex]\theta[/latex] and [latex]\phi{.}[/latex]
- 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.
47.
Draw two different angles [latex]\alpha[/latex] and [latex]\beta[/latex] in standard position whose cosine is [latex]0.3{.}[/latex]
- Use a protractor to measure [latex]\alpha[/latex] and [latex]\beta{.}[/latex]
- 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]
- Use a protractor to measure [latex]\theta[/latex] and [latex]\phi{.}[/latex]
- 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.
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.
- 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] - What happens to [latex]\tan \theta[/latex] as [latex]\theta[/latex] increases toward [latex]90°{?}[/latex]
- 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] - What happens to [latex]\tan \theta[/latex] as [latex]\theta[/latex] decreases toward [latex]90°{?}[/latex]
- What value does your calculator give for [latex]\tan 90°{?}[/latex] Why?
58.
- 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] - 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] - Plot the points from the tables and sketch a graph of [latex]f(\theta) = \tan \theta{.}[/latex]
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] |
- What happens to the slope of the line as [latex]\alpha[/latex] increases toward [latex]90°{?}[/latex]
- 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] |
- What happens to the angle of inclination as the slope increases toward infinity?
- What happens to the angle of inclination as the slope decreases toward negative infinity?