4.2 Graphs of Trigonometric Functions – Trigonometry (2024)

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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.)

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.)

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]

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]

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]

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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.

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.

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.

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.

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.

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]

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.

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]

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.

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.

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?