unit 4 revised graphs of trigonometric functions homework answers

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.

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

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?

[latex]36°{,}[/latex] 26 miles

[latex]124°{,}[/latex] 80 km

[latex]230°{,}[/latex] 120 km

[latex]318°{,}[/latex] 75 miles

[latex]285°{,}[/latex] 32 km

[latex]192°{,}[/latex] 260 miles

• 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]
• 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.
• Connect the tops of the segments to sketch a graph of [latex]y = \cos \theta{.}[/latex]
• 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]
• Sketch a graph of [latex]f(\theta) = \cos \theta[/latex] by plotting points for multiples of [latex]45°{.}[/latex]
• 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]
• Sketch a graph of [latex]f(\theta) = \sin \theta[/latex] by plotting points for multiples of [latex]30°{.}[/latex]

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]

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]\displaystyle f(\theta) = \sin \theta[/latex]
• [latex]\displaystyle f(\theta) = \cos \theta[/latex]

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

[latex]3 + f(30°)[/latex]

[latex]3 f(30°)[/latex]

[latex]4g(225°) - 1[/latex]

[latex]-4 + 2g(225°) - 1[/latex]

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

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

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

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

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.

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.

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

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

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

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

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

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

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

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

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

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

• What happens to [latex]\tan \theta[/latex] as [latex]\theta[/latex] increases toward [latex]90°{?}[/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?

• Sketch by hand a graph of [latex]y = \tan \theta[/latex] for [latex]-180° \le \theta \le 180°{.}[/latex]
• 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.

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

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

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

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

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

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

[latex](3,-5), ~\alpha = 28°[/latex]

[latex](-2,6), ~\alpha = 67°[/latex]

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

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

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.

• 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]

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.

• 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?

Trigonometry Copyright © 2024 by Bimal Kunwor; Donna Densmore; Jared Eusea; and Yi Zhen. All Rights Reserved.

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