Exercises 6.1 Arclength and Radians
Skills
- Express angles in degrees and radians #1–8, 25–32
- Sketch angles given in radians #1 and 2, 11 and 12
- Estimate angles in radians #9–10, 13–24
- Use the arclength formula #33–46
- Find coordinates of a point on a unit circle #47–52
- Calculate angular velocity and area of a sector #55–60
Problems: #4, 26, 30, 12, 10, 20, 22, 34, 36, 38, 42, 44, 50, 56, 58, 60
Exercises Homework 6.1
1.
| Radians |
[latex]0[/latex] |
[latex]\dfrac{\pi}{4}[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\dfrac{3\pi}{4}[/latex] |
[latex]\pi[/latex] |
[latex]\dfrac{5\pi}{4}[/latex] |
[latex]\dfrac{3\pi}{2}[/latex] |
[latex]\dfrac{7\pi}{4}[/latex] |
[latex]2 \pi[/latex] |
| Degrees |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
- Convert each angle to degrees.
- Sketch each angle on a circle like this one and label in radians.
2.
| Radians |
[latex]0[/latex] |
[latex]\dfrac{\pi}{6}[/latex] |
[latex]\dfrac{\pi}{3}[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\dfrac{2\pi}{3}[/latex] |
[latex]\dfrac{5\pi}{6}[/latex] |
[latex]\pi[/latex] |
[latex]\dfrac{7\pi}{6}[/latex] |
[latex]\dfrac{4\pi}{3}[/latex] |
[latex]\dfrac{3\pi}{2}[/latex] |
[latex]\dfrac{5\pi}{3}[/latex] |
[latex]\dfrac{11\pi}{6}[/latex] |
[latex]2 \pi[/latex] |
| Degrees |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
- Convert each angle to degrees.
- Sketch each angle on a circle like this one and label in radians.
Exercise Group
For Problems 3–6, express each fraction of one complete rotation in degrees and in radians.
3.
- [latex]\displaystyle \dfrac{1}{3}[/latex]
- [latex]\displaystyle \dfrac{2}{3}[/latex]
- [latex]\displaystyle \dfrac{4}{3}[/latex]
- [latex]\displaystyle \dfrac{5}{3}[/latex]
4.
- [latex]\displaystyle \dfrac{1}{5}[/latex]
- [latex]\displaystyle \dfrac{2}{5}[/latex]
- [latex]\displaystyle \dfrac{3}{5}[/latex]
- [latex]\displaystyle \dfrac{4}{5}[/latex]
5.
- [latex]\displaystyle \dfrac{1}{8}[/latex]
- [latex]\displaystyle \dfrac{3}{8}[/latex]
- [latex]\displaystyle \dfrac{5}{8}[/latex]
- [latex]\displaystyle \dfrac{7}{8}[/latex]
6.
- [latex]\displaystyle \dfrac{1}{12}[/latex]
- [latex]\displaystyle \dfrac{1}{6}[/latex]
- [latex]\displaystyle \dfrac{5}{12}[/latex]
- [latex]\displaystyle \dfrac{5}{6}[/latex]
Exercise Group
For Problems 7–8, label each angle in standard position with radian measure.
7.
Rotate counterclockwise from 0.
8.
Rotate clockwise from 0.
Exercise Group
For Problems 9–10, give a decimal approximation to hundredths for each angle in radians.
9.
- [latex]\displaystyle \dfrac{\pi}{6}[/latex]
- [latex]\displaystyle \dfrac{5\pi}{6}[/latex]
- [latex]\displaystyle \dfrac{7\pi}{6}[/latex]
- [latex]\displaystyle \dfrac{11\pi}{6}[/latex]
10.
- [latex]\displaystyle \dfrac{\pi}{4}[/latex]
- [latex]\displaystyle \dfrac{3\pi}{4}[/latex]
- [latex]\displaystyle \dfrac{5\pi}{4}[/latex]
- [latex]\displaystyle \dfrac{7\pi}{4}[/latex]
11.
Locate and label each angle from Problem 9 on the unit circle below. (The circle is marked off in tenths of a radian.)
12.
Locate and label each angle from Problem 10 on the unit circle below. (The circle is marked off in tenths of a radian.)
Exercise Group
From the list below, choose the best decimal approximation for each angle in radians in Problems 13–20. Do not use a calculator; use the fact that [latex]\pi[/latex] is a little greater than 3.
[latex]0.52,~~ 0.79,~~ 2.09,~~ 2.36,~~ 2.62,~~ 3.67,~~ 5.24,~~ 5.50[/latex]
13.
[latex]\dfrac{2\pi}{3}[/latex]
14.
[latex]\dfrac{\pi}{4}[/latex]
15.
[latex]\dfrac{5\pi}{6}[/latex]
16.
[latex]\dfrac{5\pi}{3}[/latex]
17.
[latex]\dfrac{\pi}{6}[/latex]
18.
[latex]\dfrac{7\pi}{4}[/latex]
19.
[latex]\dfrac{3\pi}{4}[/latex]
20.
[latex]\dfrac{7\pi}{6}[/latex]
Exercise Group
For Problems 21–24, say in which quadrant each angle lies.
21.
- [latex]\displaystyle \dfrac{2\pi}{3}[/latex]
- [latex]\displaystyle \dfrac{11\pi}{6}[/latex]
- [latex]\displaystyle \dfrac{-\pi}{6}[/latex]
- [latex]\displaystyle \dfrac{-7\pi}{4}[/latex]
22.
- [latex]\displaystyle \dfrac{5\pi}{4}[/latex]
- [latex]\displaystyle \dfrac{3\pi}{8}[/latex]
- [latex]\displaystyle \dfrac{-5\pi}{6}[/latex]
- [latex]\displaystyle \dfrac{-2\pi}{3}[/latex]
23.
- [latex]\displaystyle 3.5[/latex]
- [latex]\displaystyle 1.9[/latex]
- [latex]\displaystyle 0.8[/latex]
- [latex]\displaystyle 5.5[/latex]
24.
- [latex]\displaystyle 4.0[/latex]
- [latex]\displaystyle 2.6[/latex]
- [latex]\displaystyle 6.1[/latex]
- [latex]\displaystyle 1.5[/latex]
Exercise Group
For Problems 25–28, complete the table.
25.
| Radians |
[latex]\dfrac{\pi}{6}[/latex] |
[latex]\dfrac{\pi}{4}[/latex] |
[latex]\dfrac{\pi}{3}[/latex] |
| Degrees |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
26.
| Radians |
[latex]\dfrac{2\pi}{3}[/latex] |
[latex]\dfrac{3\pi}{4}[/latex] |
[latex]\dfrac{5\pi}{6}[/latex] |
| Degrees |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
27.
| Radians |
[latex]\dfrac{7\pi}{6}[/latex] |
[latex]\dfrac{5\pi}{4}[/latex] |
[latex]\dfrac{4\pi}{3}[/latex] |
| Degrees |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
28.
| Radians |
[latex]\dfrac{5\pi}{3}[/latex] |
[latex]\dfrac{7\pi}{4}[/latex] |
[latex]\dfrac{11\pi}{6}[/latex] |
| Degrees |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
Exercise Group
For Problems 29–30, convert to radians. Round to hundredths.
29.
- [latex]\displaystyle 75°[/latex]
- [latex]\displaystyle 236°[/latex]
- [latex]\displaystyle 327°[/latex]
30.
- [latex]\displaystyle 138°[/latex]
- [latex]\displaystyle 194°[/latex]
- [latex]\displaystyle 342°[/latex]
Exercise Group
For Problems 31–32, convert to degrees. Round to tenths.
31.
- [latex]\displaystyle 0.8[/latex]
- [latex]\displaystyle 3.5[/latex]
- [latex]\displaystyle 5.1[/latex]
32.
- [latex]\displaystyle 1.1[/latex]
- [latex]\displaystyle 2.6[/latex]
- [latex]\displaystyle 4.6[/latex]
Exercise Group
For Problems 33–37, use the arclength formula to answer the questions. Round answers to hundredths.
33.
Find the arclength spanned by an angle of [latex]80°[/latex] on a circle of radius 4 inches.
34.
Find the arclength spanned by an angle of [latex]200°[/latex] on a circle of radius 18 feet.
35.
Find the radius of a circle if an angle of [latex]250°[/latex] spans an arclength of 18 meters.
36.
Find the radius of a circle if an angle of [latex]20°[/latex] spans an arclength of 0.5 kilometers.
37.
Find the angle subtended by an arclength of 28 centimeters on a circle of diameter 20 centimeters.
38.
Find the angle subtended by an arclength of 1.6 yards on a circle of diameter 2 yards.
Exercise Group
For Problems 39–46, use the arclength formula to answer the questions.
39.
- Through how many radians does the minute hand of a clock sweep between 9:05 pm and 9:30 pm?
- The dial of Big Ben’s clock in London is 23 feet in diameter. How long is the arc traced by the minute hand between 9:05 pm and 9:30 pm?
40.
The largest clock ever constructed was the Floral Clock in the garden of the 1904 World’s Fair in St. Louis. The hour hand was 50 feet long, the minute hand was 75 feet long, and the radius of the clockface was 112 feet.
- If you started at the 12 and walked 500 feet clockwise around the clockface, through how many radians would you walk?
- If you started your walk at noon, how long would it take the minute hand to reach your position? How far did the tip of the minute hand move in its arc?
41.
In 1851 Jean-Bernard Foucault demonstrated the rotation of the earth with a pendulum installed in the Pantheon in Paris. Foucault’s pendulum consisted of a cannonball suspended on a 67-meter wire, and it swept out an arc of 8 meters on each swing. Through what angle did the pendulum swing? Give your answer in radians and then in degrees, rounded to the nearest hundredth.
42.
A wheel with radius 40 centimeters is rolled a distance of 1000 centimeters on a flat surface. Through what angle has the wheel rotated? Give your answer in radians and then in degrees, rounded to one decimal place.
43.
Clothes dryers draw 3.5 times as much power as washing machines, so newer machines have been engineered for greater efficiency. A vigorous spin cycle reduces the time needed for drying, and some front-loading models spin at a rate of 1500 rotations per minute.
- If the radius of the drum is 11 inches, how far do your socks travel in one minute?
- How fast are your socks traveling during the spin cycle?
44.
The Hubble telescope is in orbit around the earth at an altitude of 600 kilometers, and completes one orbit in 97 minutes.
- How far does the telescope travel in one hour? (The radius of the earth is 6400 kilometers.)
- What is the speed of the Hubble telescope?
45.
The first large windmill used to generate electricity was built in Cleveland, Ohio, in 1888. Its sails were 17 meters in diameter and moved at 10 rotations per minute. How fast did the ends of the sails travel?
46.
The largest windmill operating today has wings 54 meters in length. To be most efficient, the tips of the wings must travel at 50 meters per second. How fast must the wings rotate?
Exercise Group
For Problems 47–52, find two points on the unit circle with the given coordinate. Sketch the approximate location of the points on the circle. (Hint: what is the equation for the unit circle?)
47.
[latex]x = 0.2[/latex]
48.
[latex]x = -0.6[/latex]
49.
[latex]y = -0.35[/latex]
50.
[latex]y = 0.7[/latex]
51.
[latex]x = \dfrac{-\sqrt{3}}{2}[/latex]
52.
[latex]y = \dfrac{1}{\sqrt{2}}[/latex]
53.
- Sketch a circle of radius 4 units and mark the positions of 1, 2, 3, 4, 5, and 6 radians on the circle.
- On a circle of radius 4 feet, find the arclength determined by each angle in radians.
| [latex]\theta[/latex] |
[latex]1[/latex] |
[latex]2[/latex] |
[latex]3[/latex] |
[latex]4[/latex] |
[latex]5[/latex] |
[latex]6[/latex] |
| [latex]s[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
- Graph [latex]\theta[/latex] against [latex]s{.}[/latex] What is the slope of the graph?
- If you double the angle, what happens to the arclength? What happens if you triple [latex]\theta[/latex]?
54.
- Sketch several concentric circles with increasing radius, and draw an angle of 2 radians through all of them.
- Find the arclength determined by an angle of 2 radians on circles of given radius.
| [latex]r[/latex] |
[latex]1[/latex] |
[latex]2[/latex] |
[latex]3[/latex] |
[latex]4[/latex] |
[latex]5[/latex] |
[latex]6[/latex] |
| [latex]s[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
- Graph [latex]s[/latex] against [latex]r{.}[/latex] What is the slope of the graph?
- If you double the radius [latex]r{,}[/latex] what happens to the arclength? What happens if you triple [latex]r{?}[/latex]
55.
The angular velocity, [latex]\omega{,}[/latex] of a rotating object is given in radians per unit time. Thus, an object that rotates through [latex]\theta[/latex] radians in time [latex]t[/latex] has angular velocity given by [latex]\omega = \dfrac{\theta}{t}{.}[/latex] Find the angular velocity of the following objects.
- The London Eye Ferris wheel, which makes one revolution every 20 minutes.
- An old-fashioned long-playing record, which revolved [latex]33\frac{1}{3}[/latex] times in 60 seconds.
56.
Use the arclength formula to derive a formula relating linear velocity, [latex]v{,}[/latex] and angular velocity, [latex]\omega{.}[/latex] (See Problem 55 for the definition of angular velocity.) Start with the formula for linear velocity:
[latex]{velocity} = \dfrac{{distance}}{{time}},~~~ {or} ~~~ v = \dfrac{s}{t}[/latex]
and substitute the arclength formula for [latex]s{.}[/latex]
57.
Recall that to calculate a fraction of a revolution in degrees, we divide the angle by [latex]360°{.}[/latex] For example, [latex]90°[/latex] is [latex]\dfrac{1}{4}[/latex] of a revolution because [latex]\dfrac{90°}{360°} = \dfrac{1}{4}{.}[/latex]
- Write an expression that gives the fraction of a revolution for an angle [latex]\theta[/latex] in radians.
- Use your expression to calculate what fraction of a revolution is represented by each of the following angles: [latex]~\theta = \dfrac{3\pi}{4}, ~~ \theta = \dfrac{5\pi}{3}, ~~ \theta = \dfrac{7\pi}{6}{.}[/latex]
58.
Use your result from Problem 57a to write each statement as a mathematical formula.
- The length of the arc, [latex]s{,}[/latex] is equal to:
[latex]{(the ~~fraction ~~of ~~a ~~revolution)} \times {(circumference ~~of ~~the ~~circle)}[/latex]
- The area of the sector, [latex]A{,}[/latex] is equal to:
[latex]{(the ~~fraction ~~of ~~a ~~revolution)} \times {(area ~~of ~~the ~~circle)}[/latex]
Exercise Group
For Problems 59–60, use the formula for the area of a sector from Problem 58.
59.
60.