1.
[latex]y=4-2\sin 3x[/latex]
Chapter 7: Circular Functions
Exercises: Chapter 7 Review Problems
Chapter Review Suggested Problems
Problems: #2, 6, 12, 14, 22, 28, 30, 40
Exercise Group
For Problems 1–4, state the amplitude, period, and midline of the graph.
2.
[latex]y=1+5\cos \dfrac{x}{2}[/latex]
3.
[latex]y=2.5\cos \pi x - 2[/latex]
4.
[latex]y=0.8\sin \dfrac{\pi x}{6} + 0.3[/latex]
Exercise Group
For Problems 5–8, use transformations to sketch graphs of the functions.
5.
[latex]f(t)=2+3\cos t[/latex]
6.
[latex]g(t)=-4-2\sin t[/latex]
7.
[latex]h(w)=-4\sin \pi w[/latex]
8.
[latex]q(w)=3-\cos \dfrac{w}{2}[/latex]
Exercise Group
For Problems 9–12, write an equation for the graph using sine or cosine.
9.
10.
11.
12.
Exercise Group
For Problems 13–16, complete the table of values and sketch a graph of the function.
13.
[latex]y=\sin\left(\dfrac{x}{2}+\dfrac{\pi}{6}\right)[/latex]
- What are the period and the horizontal shift?
(Hint: Factor out [latex]\dfrac{1}{2}[/latex] from [latex]\dfrac{x}{2}+\dfrac{\pi}{6}{.}[/latex]) - Fill in the table of values.
[latex]x[/latex] [latex]\dfrac{x}{2}[/latex] [latex]\dfrac{x}{2}+\dfrac{\pi}{6}[/latex] [latex]\sin\left(\dfrac{x}{2}+\dfrac{\pi}{6}\right)[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{-\pi}{6}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]0[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{6}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{4}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{3}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{2}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{2\pi}{3}[/latex] [latex]\hphantom{0000}[/latex] - Sketch the graph.
- Solve [latex]~~\sin\left(\dfrac{x}{2}+\dfrac{\pi}{6}\right)=1,~~ {for}~~ \dfrac{-2\pi}{3} \le x \le \dfrac{2\pi}{3}[/latex]
- Solve [latex]~~\sin\left(\dfrac{x}{2}+\dfrac{\pi}{6}\right)=0,~~ {for}~~ \dfrac{-2\pi}{3} \le x \le \dfrac{2\pi}{3}[/latex]
14.
[latex]f(x)=2\cos\left(3x-\dfrac{\pi}{2}\right)+5[/latex]
- What are the midline, period, horizontal shift, and amplitude?
- Fill in the table of values.
[latex]x[/latex] [latex]3x[/latex] [latex]3x-\dfrac{\pi}{2}[/latex] [latex]\cos\left(3x-\dfrac{\pi}{2}\right)[/latex] [latex]2\cos\left(3x-\dfrac{\pi}{2}\right)+5[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]0[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{2}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\pi[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{3\pi}{2}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]2\pi[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] - Sketch the graph.
- Solve [latex]~~2\cos\left(3x-\dfrac{\pi}{2}\right)+5=7,~~ {for}~~ 0 \le x \le 2\pi[/latex]
- Solve [latex]~~2\cos\left(3x-\dfrac{\pi}{2}\right)+5=5,~~ {for}~~ 0 \le x \le 2\pi[/latex]
15.
[latex]y=20-5\cos\left(\dfrac{\pi}{30}x\right)[/latex]
- What are the midline, period, horizontal shift, and amplitude?
- Fill in the table of values.
[latex]x[/latex] [latex]\dfrac{\pi}{30}x[/latex] [latex]\cos\left(\dfrac{\pi}{30}x\right)[/latex] [latex]20-5\cos\left(\dfrac{\pi}{30}x\right)[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{-\pi}{6}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]0[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{6}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{3}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{2}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\pi[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] - Sketch the graph.
- Solve [latex]~~20-5\cos\left(\dfrac{\pi}{30}x\right)=25,~~ {for}~~ 0 \le x \le 60[/latex]
- Solve [latex]~~20-5\cos\left(\dfrac{\pi}{30}x\right)=20,~~ {for}~~ 0 \le x \le 60[/latex]
16.
[latex]y=50-50\cos(2\pi x)[/latex]
- What are the midline, period, horizontal shift, and amplitude?
- Fill in the table of values.
[latex]x[/latex] [latex]2\pi x[/latex] [latex]\cos(2\pi x)[/latex] [latex]50-50\cos(2\pi x)[/latex] [latex]\hphantom{0000}[/latex] [latex]0[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{4}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{3}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{2}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\pi[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\dfrac{-\pi}{3}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] - Sketch the graph.
- Solve [latex]~~50-50\cos(2\pi x)=50,~~ {for}~~ -1 \le x \le 1[/latex]
- Solve [latex]~~50-50\cos(2\pi x)=0,~~ {for}~~ -1 \le x \le 1[/latex]
Exercise Group
For Problems 17–18, label the scales on the axes for the graph.
17.
[latex]y=\dfrac{1}{4}\sin\left(\dfrac{x}{6}\right)+\dfrac{1}{2}[/latex]
18.
[latex]y=\dfrac{3}{2}\cos\left(\dfrac{x}{2}\right)-2[/latex]
Exercise Group
For Problems 19–20,
- Use a calculator to graph the function for [latex]0 \le x \le 2\pi{.}[/latex]
- Use the intersect feature to find all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex] Round your answers to hundredths.
19.
- [latex]\displaystyle y=-5\cos (2x-0.5)+3[/latex]
- [latex]\displaystyle -5\cos (2x-0.5)+3=-1[/latex]
20.
- [latex]\displaystyle y=2-4\sin 3(x+0.2)[/latex]
- [latex]\displaystyle 2-4\sin 3(x+0.2)=5[/latex]
Exercise Group
For Problems 21–22, write a formula for the function.
21.
The average high temperature in Phoenix, Arizona, is minimum in January at 66[latex]°[/latex] and maximum in July at 105[latex]°{.}[/latex] Write a sinusoidal function that models the average high temperature in Phoenix.
22.
The average monthly rainfall in Hawaii reaches a maximum of 3.4 inches in December and a minimum of 0.4 inches in June. Write a sinusoidal function that models the monthly rainfall in Hawaii.
Exercise Group
For Problems 23–24,
- Estimate the amplitude, period, and midline of a circular function that fits the data.
- Write a formula for the function.
23.
| [latex]x[/latex] | [latex]0[/latex] | [latex]2[/latex] | [latex]4[/latex] | [latex]6[/latex] | [latex]8[/latex] | [latex]10[/latex] | [latex]12[/latex] | [latex]14[/latex] |
| [latex]y[/latex] | [latex]12[/latex] | [latex]13.4[/latex] | [latex]16.2[/latex] | [latex]18[/latex] | [latex]17[/latex] | [latex]14.1[/latex] | [latex]12.1[/latex] | [latex]12.7[/latex] |
24.
| [latex]x[/latex] | [latex]0[/latex] | [latex]0.05[/latex] | [latex]0.1[/latex] | [latex]0.15[/latex] | [latex]0.2[/latex] | [latex]0.25[/latex] | [latex]0.3[/latex] | [latex]0.35[/latex] | [latex]0.4[/latex] |
| [latex]y[/latex] | [latex]8[/latex] | [latex]10.4[/latex] | [latex]11.8[/latex] | [latex]11.8[/latex] | [latex]10.4[/latex] | [latex]8[/latex] | [latex]5.6[/latex] | [latex]4.2[/latex] | [latex]4.2[/latex] |
Exercise Group
For Problems 25–28, give exact values for the solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex]
25.
[latex]10\sin 2\theta = -5[/latex]
26.
[latex]\sqrt{2}\cos 3\phi = 1[/latex]
27.
[latex]12\tan 4\beta = 0[/latex]
28.
[latex]2\sqrt{3}\tan 2\alpha = -6[/latex]
Exercise Group
For Problems 29–32, find all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex] Round your answers to three decimal places.
29.
[latex]5\tan 3x+2 = 3[/latex]
30.
[latex]-8\sin 2t - 4 = 3[/latex]
31.
[latex]2.8 - 3.6\cos 2s = 5.2[/latex]
32.
[latex]6.7 \tan 3u + 1.2 = 28[/latex]
Exercise Group
For Problems 33–36, use a substitution to find exact values for all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex]
33.
[latex]2\cos\left(2\phi - \dfrac{\pi}{4}\right)=\sqrt{3}[/latex]
34.
[latex]3\sin (3z+\pi) + 2 = -1[/latex]
35.
[latex]-4\sin\left(\dfrac{t}{2}+\dfrac{\pi}{8}\right)=\sqrt{8}[/latex]
36.
[latex]7\cos\left(\dfrac{w}{2}-\dfrac{\pi}{3}\right)=-3.5[/latex]
Exercise Group
For Problems 37–40, use a substitution to find all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex] Round your answers to hundredths.
37.
[latex]0.4\tan(3x+0.2)=1.6[/latex]
38.
[latex]15\tan\left(1.4s-2\right)=20[/latex]
39.
[latex]8\sin\left(\dfrac{\pi t}{6}-\dfrac{\pi}{12}\right) = 6[/latex]
40.
[latex]12\cos\left(\dfrac{\pi t}{2}-\dfrac{3\pi}{5}\right) = 5[/latex]