1.
How many degrees are in each angle?
- [latex]\dfrac{3}{5}[/latex] of one rotation
- [latex]\dfrac{3}{10}[/latex] of one rotation
- [latex]\dfrac{4}{3}[/latex] of one rotation
- [latex]\dfrac{8}{3}[/latex] of one rotation
Chapter 4: Trig Functions
Exercises: 4.1 Angles and Rotation
Skills
Practice each skill in the Homework Problems listed:
- Use angles to represent rotations
- Sketch angles in standard position
- Find coterminal angles
- Find and use reference angles
- Find trigonometric ratios for the special angles
- Solve equations
Suggested Problems
Problems: #2, 4, 10, 16, 28, 30, 36, 42, 58, 52, 54, 64, 70, 78
Exercises Homework 4.1
2.
How many degrees are in each angle?
- [latex]\dfrac{5}{6}[/latex] of one rotation
- [latex]\dfrac{3}{8}[/latex] of one rotation
- [latex]\dfrac{7}{4}[/latex] of one rotation
- [latex]\dfrac{7}{12}[/latex] of one rotation
3.
What fraction of a complete rotation is represented by each angle?
- [latex]45°[/latex]
- [latex]300°[/latex]
- [latex]540°[/latex]
- [latex]420°[/latex]
4.
What fraction of a complete rotation is represented by each angle?
- [latex]60°[/latex]
- [latex]240°[/latex]
- [latex]450°[/latex]
- [latex]150°[/latex]
5.
- Through what angle does the hour hand of a clock rotate between 2 pm and 10 pm?
- Through what angle does the hour hand of a clock rotate between 2 am and 10 pm?
6.
- Through what angle does the minute hand of a clock rotate between 3:25 am and 3:50 am?
- Through what angle does the minute hand of a clock rotate between 4:10 pm and 6:25 pm?
Exercise Group
For Problems 7–12, calculate the degree measure of the unknown angle and sketch the angle in standard position.
7.
8.
9.
10.
11.
12.
Exercise Group
For Problems 13–18, find two angles, one positive and one negative, that are coterminal with the given angle.
13.
[latex]40°[/latex]
14.
[latex]160°[/latex]
15.
[latex]215°[/latex]
16.
[latex]250°[/latex]
17.
[latex]305°[/latex]
18.
[latex]340°[/latex]
Exercise Group
For Problems 19–24, find a positive angle between [latex]0°[/latex] and [latex]360°[/latex] that is coterminal with the given angle.
19.
[latex]-65°[/latex]
20.
[latex]-140°[/latex]
21.
[latex]-290°[/latex]
22.
[latex]-325°[/latex]
23.
[latex]-405°[/latex]
24.
[latex]-750°[/latex]
Exercise Group
For Problems 25–30, find the reference angle. Make a sketch showing the angle, the reference angle, and the reference triangle.
25.
[latex]100°[/latex]
26.
[latex]125°[/latex]
27.
[latex]216°[/latex]
28.
[latex]242°[/latex]
29.
[latex]297°[/latex]
30.
[latex]336°[/latex]
Exercise Group
For Problems 31–36, find three angles between [latex]90°[/latex] and [latex]360°[/latex] with the given reference angle, and sketch all four angles on the same grid.
31.
[latex]15°[/latex]
32.
[latex]26°[/latex]
33.
[latex]40°[/latex]
34.
[latex]50°[/latex]
35.
[latex]68°[/latex]
36.
[latex]75°[/latex]
Exercise Group
For Problems 37–44, use the values given below to find the trigonometric ratio. Do not use a calculator!
[latex]\cos 23° = 0.9205~~~~~~\sin 46° = 0.7193~~~~~~\tan 78° = 4.7046[/latex]
37.
[latex]\cos 157°[/latex]
38.
[latex]\sin 226°[/latex]
39.
[latex]\sin 314°[/latex]
40.
[latex]\cos 203°[/latex]
41.
[latex]\tan 258°[/latex]
42.
[latex]\tan 282°[/latex]
43.
[latex]\sin (-134°)[/latex]
44.
[latex]\cos (-383°)[/latex]
45.
On the circle in the figure, all angles are shown in standard position. Find the measure in degrees of the angles labeled (a)-(i).
46.
Find the reference angle for each of your answers in Problem 45.
47.
- Draw three angles, one in each quadrant except the first, whose reference angle is [latex]60°{.}[/latex]
- Find exact values for the sine, cosine, and tangent of each of the angles in part (a).
48.
- Draw three angles, one in each quadrant except the first, whose reference angle is [latex]30°{.}[/latex]
- Find exact values for the sine, cosine, and tangent of each of the angles in part (a).
49.
- Draw three angles, one in each quadrant except the first, whose reference angle is [latex]45°{.}[/latex]
- Find exact values for the sine, cosine, and tangent of each of the angles in part (a).
50.
Complete the table with exact values.
| [latex]\theta[/latex] | [latex]30°[/latex] | [latex]45°[/latex] | [latex]60°[/latex] | [latex]120°[/latex] | [latex]135°[/latex] | [latex]150°[/latex] | [latex]210°[/latex] | [latex]225°[/latex] | [latex]240°[/latex] | [latex]300°[/latex] | [latex]315°[/latex] | [latex]330°[/latex] |
| [latex]\cos \theta[/latex] | ||||||||||||
| [latex]\sin \theta[/latex] | ||||||||||||
| [latex]\tan \theta[/latex] |
51.
In which two quadrants is the statement true?
- The sine is negative.
- The cosine is negative.
- The tangent is positive.
52.
Find all angles between [latex]0°[/latex] and [latex]360°[/latex] for which the statement is true.
- [latex]\cos \theta = -1[/latex]
- [latex]\sin \theta = -1[/latex]
- [latex]\tan \theta = -1[/latex]
53.
- Find two angles, [latex]0 \le \theta \le 360°{,}[/latex] with [latex]\sin \theta = 0{.}[/latex]
- Find two angles, [latex]0 \le \theta \le 360°{,}[/latex] with [latex]\cos \theta = 0{.}[/latex]
54.
- Find two angles, [latex]0 \le \theta \le 360°{,}[/latex] with [latex]\sin \theta = \cos \theta{.}[/latex]
- Find two angles, [latex]0 \le \theta \le 360°{,}[/latex] with [latex]\sin \theta = -\cos \theta{.}[/latex]
Exercise Group
For Problems 55–60, find a second angle between [latex]0°[/latex] and [latex]360°[/latex] with the given trigonometric ratio.
55.
[latex]\sin 75°[/latex]
56.
[latex]\cos 32°[/latex]
57.
[latex]\tan 84°[/latex]
58.
[latex]\sin 16°[/latex]
59.
[latex]\cos 47°[/latex]
60.
[latex]\tan 56°[/latex]
Exercise Group
For Problems 61–66, find all solutions between [latex]0°[/latex] and [latex]360°{.}[/latex] Round to the nearest degree.
61.
[latex]\tan \theta = 8.1443[/latex]
62.
[latex]\sin \theta = 0.7880[/latex]
63.
[latex]\cos \theta = 0.9205[/latex]
64.
[latex]\tan \theta = -3.4874[/latex]
65.
[latex]\sin \theta = -0.9962[/latex]
66.
[latex]\cos \theta = -0.0349[/latex]
Exercise Group
For Problems 67–72, find exact values for all solutions between [latex]0°[/latex] and [latex]360°{.}[/latex]
67.
[latex]\cos \theta = -\cos 24°[/latex]
68.
[latex]\tan \theta = -\tan 9°[/latex]
69.
[latex]\sin \theta =-\sin 66°[/latex]
70.
[latex]\cos \theta = -\cos 78°[/latex]
71.
[latex]\tan \theta = -\tan 31°[/latex]
72.
[latex]\sin \theta = -\sin 42°[/latex]
Exercise Group
For Problems 73–78, use similar triangles to find the coordinates of the point on the circle.
73.
74.
75.
76.
77.
78.
Exercise Group
For Problems 79–82,
- Use the grid to estimate the coordinates of the point [latex]P[/latex] on the unit circle.
- Use a calculator to find the coordinates of the point [latex]P{.}[/latex] Round to thousandths.
- Estimate the coordinates of the point [latex]Q[/latex] on the circle of radius 2 and verify with a calculator.
79.
80.
81.
82.
83.
Explain why the definitions of the trigonometric ratios for a third-quadrant angle (between [latex]180°[/latex] and [latex]270°[/latex]) are independent of the point [latex]P[/latex] chosen on the terminal side. Illustrate with a figure.
84.
Explain why the definitions of the trigonometric ratios for a fourth-quadrant angle (between [latex]270°[/latex] and [latex]360°[/latex]) are independent of the point [latex]P[/latex] chosen on the terminal side. Illustrate with a figure.