1.
The London Eye, the world’s largest Ferris wheel, completes one revolution every 30 minutes. By how many degrees will it rotate in 1 minute?
Chapter 4: Trig Functions
Exercises: Chapter 4 Review Problems
Chapter Review Suggested Problems
Problems: # 4, 6, 14, 26, 30, 34, 36, 42, 52, 58, 62
Exercises for Chapter 4 Review
Chapter 4 Review Problems
2.
The London Eye in Problem 1 has 32 cabins evenly spaced along the wheel. If the cabins are numbered consecutively from 1 to 32, what is the angular separation between cabins number 1 and number 15?
Exercise Group
For Problems 3–4, find two angles, one positive and one negative, that are coterminal with the given angle.
3.
- [latex]\displaystyle 510°[/latex]
- [latex]\displaystyle 600°[/latex]
- [latex]\displaystyle -200°[/latex]
- [latex]\displaystyle -700°[/latex]
4.
- [latex]\displaystyle -380°[/latex]
- [latex]\displaystyle -423°[/latex]
- [latex]\displaystyle 187°[/latex]
- [latex]\displaystyle 1000°[/latex]
Exercise Group
For the angles in Problems 5–6, state the corresponding quadrant and reference angle. Give three other angles with the same reference angle, one for each of the other three quadrants. Sketch all four angles.
5.
- [latex]\displaystyle -300°[/latex]
- [latex]\displaystyle -25°[/latex]
- [latex]\displaystyle 100°[/latex]
- [latex]\displaystyle 250°[/latex]
6.
- [latex]\displaystyle 430°[/latex]
- [latex]\displaystyle -590°[/latex]
- [latex]\displaystyle -95°[/latex]
- [latex]\displaystyle 1050°[/latex]
7.
Let [latex]\widetilde{\theta} = f(\theta)[/latex] be the function that gives the reference angle of [latex]\theta{.}[/latex] For example, [latex]f(110°) = 70°[/latex] because the reference angle for [latex]110°[/latex] is [latex]70°{.}[/latex]
- Fill in the table of values.
[latex]\theta[/latex] [latex]0°[/latex] [latex]30°[/latex] [latex]60°[/latex] [latex]90°[/latex] [latex]120°[/latex] [latex]150°[/latex] [latex]180°[/latex] [latex]210°[/latex] [latex]240°[/latex] [latex]270°[/latex] [latex]300°[/latex] [latex]330°[/latex] [latex]360°[/latex] [latex]f(\theta)[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] - Choose appropriate scales for the axes and graph the function for [latex]-360° \le \theta \le 360°{.}[/latex]
8.
Let [latex]\widetilde{\theta} = f(\theta)[/latex] be the function that gives the reference angle of [latex]\theta{.}[/latex] (See Problem 7.) Is [latex]f[/latex] a periodic function? If so, give its period, midline, and amplitude. If not, explain why not.
Exercise Group
For Problems 9–20, solve the equation exactly for [latex]0° \le \theta \le 360°{.}[/latex]
9.
[latex]\sin \theta = \dfrac{-1}{2}[/latex]
10.
[latex]\cos \theta = \dfrac{-1}{\sqrt{2}}[/latex]
11.
[latex]2\cos \theta + 1 = 0[/latex]
12.
[latex]5\sin \theta + 5 = 0[/latex]
13.
[latex]\tan \theta - 1 = 0[/latex]
14.
[latex]\sqrt{3} + 3\tan \theta = 0[/latex]
15.
[latex]\cos \theta = \cos(-23°)[/latex]
16.
[latex]\sin \theta = \sin(370°)[/latex]
17.
[latex]\tan \theta = \tan 432°[/latex]
18.
[latex]\tan \theta = \tan (-6°)[/latex]
19.
[latex]\sin \theta + \sin 83° = 0[/latex]
20.
[latex]\cos \theta + \cos 429° = 0[/latex]
Exercise Group
For Problems 21–26, solve the equation for [latex]0° \le \theta \le 360°{.}[/latex] Round your answers to two decimal places.
21.
[latex]3\sin \theta + 2 = 0[/latex]
22.
[latex]5\cos \theta + 4 = 0[/latex]
23.
[latex]\dfrac{2}{3}\tan \theta + 1 = 0[/latex]
24.
[latex]-4\tan \theta + 12 = 0[/latex]
25.
[latex]4 = 8\cos \theta + 9[/latex]
26.
[latex]-3 = 6\sin \theta -5[/latex]
Exercise Group
For Problems 27–32, find the coordinates of the point where the terminal side of angle [latex]\theta[/latex] in standard position intersects the circle of radius [latex]r[/latex] centered at the origin. Round your answers to two decimal places.
27.
[latex]\theta = 193°,~ r = 10[/latex]
28.
[latex]\theta = -12°,~ r = 20[/latex]
29.
[latex]\theta = 92°,~ r = 8[/latex]
30.
[latex]\theta = 403°,~ r = 6[/latex]
31.
[latex]\theta = -341°,~ r = 3[/latex]
32.
[latex]\theta = -107°,~ r = 20[/latex]
33.
If you follow a bearing of [latex]190°[/latex] for 10 miles, how far south and how far west are you from your starting point? Round to two decimal places.
34.
A child releases a balloon, which then follows a bearing of [latex]84°[/latex] for 150 meters. How far east and how far north is the balloon from where it was released? Round to the nearest meter.
Exercise Group
For Problems 35–38, write the equation of a sine or cosine function with the given properties and sketch the graph, including at least one period.
35.
Amplitude 7, midline [latex]y = 4{,}[/latex] period 2, [latex]y[/latex]-intercept 4
36.
Amplitude 100, midline [latex]y = 50{,}[/latex] period 12, [latex]y[/latex]-intercept 100
37.
Maximum point at [latex](90°, 24)[/latex] and minimum point at [latex](270°, 10)[/latex]
38.
Horizontal intercept at [latex]180°{,}[/latex] maximum points at [latex](0°, 5)[/latex] and [latex](360°, 5)[/latex]
Exercise Group
For Problems 39–42, evaluate the expression for [latex]f(\theta) = \sin \theta[/latex] and [latex]g(\theta) = \cos \theta{.}[/latex]
39.
[latex]2f(\theta)g(\theta)[/latex] for [latex]\theta = 30°[/latex]
40.
[latex]f(\dfrac{\theta}{2})[/latex] for [latex]\theta = 120°[/latex]
41.
[latex]g(4\theta) - f(2\theta)[/latex] for [latex]\theta = 15°[/latex]
42.
[latex]\dfrac{1 - g(2\theta)}{2}[/latex] for [latex]\theta = 45°[/latex]
Exercise Group
For Problems 43–46, write an equation for the given graph, and give the exact coordinates of the labeled points.
43.
44.
45.
46.
47.
Every 24 hours Delbert takes 50 mg of a therapeutic drug. The level of that drug in Delbert’s bloodstream immediately jumps to its peak level of 60 mg, but the level diminishes to its lowest level of 10 mg just before the next dose.
- Sketch a graph of [latex]d(t){,}[/latex] the amount of the drug in Delbert’s bloodstream after one of his doses, Show at least three doses.
- What is the period of [latex]d(t){?}[/latex]
48.
A water fountain has water trickling into a container, but once the container is full, it tilts and the water pours quickly out. Then the container tilts back and starts to fill again. The container is filled 5 times every minute.
- Sketch a graph of [latex]h(t){,}[/latex] the height of water in the container at time Show at least two complete fill-and-empty cycles.
- What is the period of [latex]h(t){?}[/latex]
49.
Henry is watching Billie ride a carousel. He stands 2 meters from the carousel, which has a diameter of 10 meters. He notices that she passes by him three times each minute.
- Sketch a graph of [latex]f(t){,}[/latex] the distance between Henry and Billie at time [latex]t{.}[/latex] Show at least two complete circuits.
- What is the period of [latex]f(t){?}[/latex]
50.
An ant walks at constant unit speed (1 unit of distance per second) along the triangle with vertices at [latex](0,0),~(1,1),[/latex] and [latex](0,1){.}[/latex]
- How far does the ant need to walk to get from [latex](0,0)[/latex] to [latex](1,1){?}[/latex] From [latex](1,1)[/latex] to [latex](0,1){?}[/latex] From [latex](0,1)[/latex] to [latex](0,0){?}[/latex]
- Sketch a graph of [latex]g(t){,}[/latex] the ant’s [latex]y[/latex]-coordinate at time [latex]t{.}[/latex] Show at least two circuits around the triangle.
- What is the period of [latex]g(t){?}[/latex]
Exercise Group
For Problems 51–54,
- Graph the function.
- State the amplitude, period, and midline of the function.
51.
[latex]y = 4 + 2 \cos \theta[/latex]
52.
[latex]y = -1 + 3 \cos \theta[/latex]
53.
[latex]y = 1.5 + 3.5 \sin 2\theta[/latex]
54.
[latex]y = 1.6 + 1.4 \sin (0.5\theta)[/latex]
Exercise Group
For Problems 55–58, find the angle of inclination of the line.
55.
[latex]y = \dfrac{\sqrt{3}}{3} x + 1[/latex]
56.
[latex]y = - x - 11[/latex]
57.
[latex]y = 100 - 28 x[/latex]
58.
[latex]y = - 3.7 + 1.4x[/latex]
Exercise Group
For Problems 59–62, find an equation for the line passing through the given point with angle of inclination [latex]\alpha{.}[/latex]
59.
[latex](0,2){,}[/latex] [latex]~ \alpha = 45°[/latex]
60.
[latex](4,0){,}[/latex] [latex]~ \alpha = 135°[/latex]
61.
[latex](3,-4){,}[/latex] [latex]~ \alpha = 120°[/latex]
62.
[latex](-7,2){,}[/latex] [latex]~ \alpha = 60°[/latex]
63.
Sketch the graphs of [latex]y = \tan \theta[/latex] and [latex]y = \cos \theta[/latex] on the same grid for [latex]-180° \lt \theta \lt 180°{.}[/latex] How are the [latex]\theta[/latex]-intercepts of the graph of [latex]y = \cos \theta[/latex] related to the graph of [latex]y = \tan \theta[/latex]?
64.
Sketch the graphs of [latex]y = \tan \theta[/latex] and [latex]y = \sin \theta[/latex] on the same grid for [latex]-180° \lt \theta \lt 180°{.}[/latex] How are the [latex]\theta[/latex]-intercepts of the graph of [latex]y = \sin \theta[/latex] related to the graph of [latex]y = \tan \theta[/latex]?