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# 5.2.2E: Angles (Exercises)

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Section 5.2 exercise

1. Indicate each angle on a circle: 30$$\mathrm{{}^\circ}$$, 300$$\mathrm{{}^\circ}$$, -135$$\mathrm{{}^\circ}$$, 70$$\mathrm{{}^\circ}$$, $$\dfrac{2\pi }{3}$$, $$\dfrac{7\pi }{4}$$
2. Indicate each angle on a circle: 30$$\mathrm{{}^\circ}$$, 315$$\mathrm{{}^\circ}$$, -135$$\mathrm{{}^\circ}$$, 80$$\mathrm{{}^\circ}$$, $$\dfrac{7\pi }{6}$$, $$\dfrac{3\pi }{4}$$
3. Convert the angle 180$$\mathrm{{}^\circ}$$ to radians.
4. Convert the angle 30$$\mathrm{{}^\circ}$$ to radians.
5. Convert the angle $$\dfrac{5\pi }{6}$$ from radians to degrees.
6. Convert the angle $$\dfrac{11\; \pi }{6}$$ from radians to degrees.
7. Find the angle between 0$$\mathrm{{}^\circ}$$ and 360$$\mathrm{{}^\circ}$$ that is coterminal with a 685$$\mathrm{{}^\circ}$$ angle.
8. Find the angle between 0$$\mathrm{{}^\circ}$$ and 360$$\mathrm{{}^\circ}$$ that is coterminal with a 451$$\mathrm{{}^\circ}$$ angle.
9. Find the angle between 0$$\mathrm{{}^\circ}$$ and 360$$\mathrm{{}^\circ}$$ that is coterminal with a -1746$$\mathrm{{}^\circ}$$ angle.
10. Find the angle between 0$$\mathrm{{}^\circ}$$ and 360$$\mathrm{{}^\circ}$$ that is coterminal with a -1400$$\mathrm{{}^\circ}$$ angle.
11. Find the angle between 0 and 2$$\pi$$ in radians that is coterminal with the angle $$\dfrac{26\; \pi }{9}$$.
12. Find the angle between 0 and 2$$\pi$$ in radians that is coterminal with the angle $$\dfrac{17\; \pi }{3}$$.
13. Find the angle between 0 and 2$$\pi$$ in radians that is coterminal with the angle $$-\dfrac{3\; \pi }{2}$$.
14. Find the angle between 0 and 2$$\pi$$ in radians that is coterminal with the angle $$-\dfrac{7\; \pi }{6}$$.
15. On a circle of radius 7 miles, find the length of the arc that subtends a central angle of 5 radians.
16. On a circle of radius 6 feet, find the length of the arc that subtends a central angle of 1 radian.
17. On a circle of radius 12 cm, find the length of the arc that subtends a central angle of 120 degrees.
18. On a circle of radius 9 miles, find the length of the arc that subtends a central angle of 800 degrees.
19. Find the distance along an arc on the surface of the Earth that subtends a central angle of 5 minutes (1 minute = 1/60 degree). The radius of the Earth is 3960 miles.
20. Find the distance along an arc on the surface of the Earth that subtends a central angle of 7 minutes (1 minute = 1/60 degree). The radius of the Earth is 3960 miles.
21. On a circle of radius 6 feet, what angle in degrees would subtend an arc of length 3 feet?
22. On a circle of radius 5 feet, what angle in degrees would subtend an arc of length 2 feet?
23. A sector of a circle has a central angle of 45$$\mathrm{{}^\circ}$$. Find the area of the sector if the radius of the circle is 6 cm.
24. A sector of a circle has a central angle of 30$$\mathrm{{}^\circ}$$. Find the area of the sector if the radius of the circle is 20 cm.
25. A truck with 32-in.-diameter wheels is traveling at 60 mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?
26. A bicycle with 24-in.-diameter wheels is traveling at 15 mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?
27. A wheel of radius 8 in. is rotating 15$$\mathrm{{}^\circ}$$/sec. What is the linear speed v, the angular speed in RPM, and the angular speed in rad/sec?
28. A wheel of radius 14 in. is rotating 0.5 rad/sec. What is the linear speed v, the angular speed in RPM, and the angular speed in deg/sec?
29. A CD has diameter of 120 millimeters. When playing audio, the angular speed varies to keep the linear speed constant where the disc is being read. When reading along the outer edge of the disc, the angular speed is about 200 RPM (revolutions per minute). Find the linear speed.
30. When being burned in a writable CD-R drive, the angular speed of a CD is often much faster than when playing audio, but the angular speed still varies to keep the linear speed constant where the disc is being written. When writing along the outer edge of the disc, the angular speed of one drive is about 4800 RPM (revolutions per minute). Find the linear speed.
31. You are standing on the equator of the Earth (radius 3960 miles). What is your linear and angular speed?
32. The restaurant in the Space Needle in Seattle rotates at the rate of one revolution every 47 minutes. [UW]
33. Through how many radians does it turn in 100 minutes?
34. How long does it take the restaurant to rotate through 4 radians?
35. How far does a person sitting by the window move in 100 minutes if the radius of the restaurant is 21 meters?

1. 3. $$\pi$$

5. $$150^{\circ}$$

7. $$325^{\circ}$$

9. $$54^{\circ}$$

11. $$\dfrac{8\pi}{9}$$

13. $$\dfrac{\pi}{2}$$

15. 35 miles

17. $$8\pi$$ cm

19. 5.7596 miles

21. $$28.6479^{\circ}$$

23. $$14.1372 \text{cm}^2$$

27. 2.094 in/sec, $$\pi$$/12 rad/sec, 2.5 RPM
31. Angular speed: $$\pi$$/12 rad/hr. Linear speed: 1036.73 miles/hr