1.
[latex]\sin 4x = -1[/latex]
Chapter 7: Circular Functions
Exercises 7.3 Solving Equations
Skills
- Find exact solutions to equations of the form [latex]\sin nx = k[/latex] #1–10
- Find all solutions between [latex]0[/latex] and [latex]2\pi[/latex] #11–16
- Use a substitution to solve trigonometric equations #17–28
- Write expressions for exact solutions #29–42
- Solve problems involving trigonometric models #43–46
Suggested Problems
Problems: # 4, 10, 12, 26, 34, 42
Exercises Homework 7.3
Exercise Group
For Problems 1–10,
- Use a graph to estimate all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex]
- Give exact values for the solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex]
2.
[latex]\cos 3t = 0[/latex]
3.
[latex]5\tan 2q = 0[/latex]
4.
[latex]6\sin 4w = -3\sqrt{2}[/latex]
5.
[latex]4\cos 3\phi = -2[/latex]
6.
[latex]\sqrt{3}\tan 2\alpha = 3[/latex]
7.
[latex]2\sin 2\beta = 1[/latex]
8.
[latex]-6\cos 2\theta = 6[/latex]
9.
[latex]3\tan 3w = \sqrt{3}[/latex]
10.
[latex]2\tan 3u = -2[/latex]
Exercise Group
For Problems 11–20, find all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex] Round your answers to three decimal places.
11.
[latex]9\cos 2\theta + 1 = 6[/latex]
12.
[latex]7\cos 2t-3=2[/latex]
13.
[latex]8\tan 4t+1=-11[/latex]
14.
[latex]3=3\tan 4x+4[/latex]
15.
[latex]5\sin 3\theta -3=-4[/latex]
16.
[latex]150\sin 3s = 27[/latex]
17.
[latex]6\cos 2r+2=3[/latex]
18.
[latex]2-8\cos 3t=-4[/latex]
19.
[latex]\dfrac{5}{7} \tan \pi x +11=11[/latex]
20.
[latex]2\tan 2\pi \beta + 5 = 3[/latex]
Exercise Group
For Problems 21–28, use a substitution to find exact values for all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex]
21.
[latex]2-\tan\left(2x-\dfrac{\pi}{3}\right)=2[/latex]
22.
[latex]2\cos\left(3t+\dfrac{\pi}{4}\right)=\sqrt{3}[/latex]
23.
[latex]6\cos\left(3\theta-\dfrac{\pi}{2}\right) = -3\sqrt{2}[/latex]
24.
[latex]8\sin \left(2\theta - \dfrac{\pi}{6}\right)=-4[/latex]
25.
[latex]7\sin \left(\dfrac{\phi}{2}+\dfrac{3\pi}{4}\right)+3=-4[/latex]
26.
[latex]3\tan\left(\dfrac{w}{2}+\dfrac{\pi}{4}\right)+4=1[/latex]
27.
[latex]160\sin(\pi \phi -1)+10=90[/latex]
28.
[latex]200\sin \left(\pi t +6\right)-10=-110[/latex]
Exercise Group
For Problems 29–42, use a substitution to find all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex] Round your answers to hundredths.
29.
[latex]16\cos(3t-1)+4=-8[/latex]
30.
[latex]3-5\cos (2\phi -1)=6[/latex]
31.
[latex]23-24\tan(\pi x+2)=17[/latex]
32.
[latex]14\tan (\pi \beta -4)+31=10[/latex]
33.
[latex]120\sin\left(\dfrac{\pi}{3}(t-0.2)\right)+21=-3[/latex]
34.
[latex]9\sin \left(\dfrac{\pi}{2}(t-1)\right)+5=-1[/latex]
35.
[latex]5\sin\left(3w-\dfrac{\pi}{3}\right)+1=4[/latex]
36.
[latex]8\tan \left(4t-\dfrac{\pi}{3}\right)-24=1[/latex]
37.
[latex]16\cos\left(\dfrac{\pi}{2}(t+0.3)\right)-7=5[/latex]
38.
[latex]5\cos \left(\dfrac{\pi}{4}\left(t+\dfrac{1}{4}\right)\right)+3=2[/latex]
39.
[latex]6\tan\left(\dfrac{\pi}{3}(\theta - 1)\right)+4=5[/latex]
40.
[latex]1.5\sin \left(\dfrac{\pi}{2}(\alpha + 0.1)\right)+0.4=0.1[/latex]
41.
[latex]5-3\cos\left(\dfrac{\pi}{6}(w+0.1)\right)=4[/latex]
42.
[latex]0.34\cos (2\pi(\alpha-0.2))=0.085[/latex]
43.
The population of deer in Marquette County over the course of a typical year can be approximated by a sinusoidal function. The population reached a maximum of 50,000 deer on September 1 and a minimum of 42,000 deer on March 1.
- Write a formula for the function [latex]P(t)[/latex] that gives the deer population on the first of each month, where [latex]t=0[/latex] is September 1.
- When is the deer population 45,000? Give exact expressions and approximations rounded to two decimal places.
- Graph your function over one period and label the points that correspond to a deer population of 45,000. Is the population greater or less than 45,000 between the two solutions?
44.
The percent of the moon visible from earth is a sinusoidal function ranging from 0% to 100%, with a period of 29.5 days.
- Write a formula for the function [latex]f(t)[/latex] that gives the percent of the moon that is visible, if a new moon (0% visible) occurs at [latex]t=0[/latex] days.
- When is 25% of the moon visible? Give approximations rounded to two decimal places.
- Graph your function over one period, and label the points that correspond to a quarter moon. Is more or less than 25% of the moon visible between the two solutions you found in part (b)?
45.
A Ferris wheel has a diameter of 20 meters and completes one revolution every 60 seconds. Delbert is at the lowest position of the Ferris wheel, 1 meter above ground, when [latex]t=0[/latex] seconds.
- Write a formula for the function [latex]h(t)[/latex] that gives Delbert’s altitude in meters after [latex]t[/latex] seconds.
- When is Delbert at an altitude of 18 meters during his first revolution? Give exact expressions and approximations rounded to two decimal places.
- Graph your function over one period, labeling the points that correspond to an altitude of 18 meters. Is Delbert above or below 18 meters between the two solutions you found in part (b)?
46.
High tides occur every 12.2 hours at Point Lookout. The depth of the water at the end of David’s dock is 2.6 meters at high tide and 1.8 meters at low tide.
- Write a formula for the function [latex]d(t)[/latex] that gives the depth of the water [latex]t[/latex] hours after last night’s high tide.
- When is the water at the end of the dock 2 meters deep? Give approximations rounded to two decimal places.
- Graph your function over one period, labeling the points that correspond to a depth of 2 meters. Is the water depth greater or less than 2 meters between the two solutions you found in part (b)?