1.
[latex]y=4-2\sin 3x[/latex]
Chapter 7: Circular Functions
Problems: #2, 6, 12, 14, 22, 28, 30, 40
For Problems 1–4, state the amplitude, period, and midline of the graph.
[latex]y=4-2\sin 3x[/latex]
[latex]y=1+5\cos \dfrac{x}{2}[/latex]
[latex]y=2.5\cos \pi x - 2[/latex]
[latex]y=0.8\sin \dfrac{\pi x}{6} + 0.3[/latex]
For Problems 5–8, use transformations to sketch graphs of the functions.
[latex]f(t)=2+3\cos t[/latex]
[latex]g(t)=-4-2\sin t[/latex]
[latex]h(w)=-4\sin \pi w[/latex]
[latex]q(w)=3-\cos \dfrac{w}{2}[/latex]
For Problems 9–12, write an equation for the graph using sine or cosine.
For Problems 13–16, complete the table of values and sketch a graph of the function.
[latex]y=\sin\left(\dfrac{x}{2}+\dfrac{\pi}{6}\right)[/latex]
| [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] |
[latex]f(x)=2\cos\left(3x-\dfrac{\pi}{2}\right)+5[/latex]
| [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] |
[latex]y=20-5\cos\left(\dfrac{\pi}{30}x\right)[/latex]
| [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] |
[latex]y=50-50\cos(2\pi x)[/latex]
| [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] |
For Problems 17–18, label the scales on the axes for the graph.
[latex]y=\dfrac{1}{4}\sin\left(\dfrac{x}{6}\right)+\dfrac{1}{2}[/latex]
[latex]y=\dfrac{3}{2}\cos\left(\dfrac{x}{2}\right)-2[/latex]
For Problems 19–20,
For Problems 21–22, write a formula for the function.
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.
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.
For Problems 23–24,
| [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] |
| [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] |
For Problems 25–28, give exact values for the solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex]
[latex]10\sin 2\theta = -5[/latex]
[latex]\sqrt{2}\cos 3\phi = 1[/latex]
[latex]12\tan 4\beta = 0[/latex]
[latex]2\sqrt{3}\tan 2\alpha = -6[/latex]
For Problems 29–32, find all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex] Round your answers to three decimal places.
[latex]5\tan 3x+2 = 3[/latex]
[latex]-8\sin 2t - 4 = 3[/latex]
[latex]2.8 - 3.6\cos 2s = 5.2[/latex]
[latex]6.7 \tan 3u + 1.2 = 28[/latex]
For Problems 33–36, use a substitution to find exact values for all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex]
[latex]2\cos\left(2\phi - \dfrac{\pi}{4}\right)=\sqrt{3}[/latex]
[latex]3\sin (3z+\pi) + 2 = -1[/latex]
[latex]-4\sin\left(\dfrac{t}{2}+\dfrac{\pi}{8}\right)=\sqrt{8}[/latex]
[latex]7\cos\left(\dfrac{w}{2}-\dfrac{\pi}{3}\right)=-3.5[/latex]
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.
[latex]0.4\tan(3x+0.2)=1.6[/latex]
[latex]15\tan\left(1.4s-2\right)=20[/latex]
[latex]8\sin\left(\dfrac{\pi t}{6}-\dfrac{\pi}{12}\right) = 6[/latex]
[latex]12\cos\left(\dfrac{\pi t}{2}-\dfrac{3\pi}{5}\right) = 5[/latex]