Q: Which formula gives the area $$A of a $$sector of a circle of radius $$r$$ formed by a central angle $$\theta (in $$radians)?

$$A = \frac{1}{2} r^2$ \theta$$

Q: A water sprinkler sprays water over a distance of 10 meters while rotating through an angle of $$120^\circ. $$What is the area of the lawn that receives water? (Use $$A = \frac{1}{2} r^2$ \theta$$ and convert angle to radians.)

$$\frac{1}{2} \times 10^2$ \times \frac{2\pi}{3} = 100\pi/3$$ square meters

Q: A child spins a rock at the end of a 2-foot rope at 180 revolutions per minute. What is the linear speed of the rock when released? (Use $$v = r\omega$$ and $$\omega in $$radians per minute.)

$$v = 2 \times 180 \times 2\pi = 720\pi$$ feet/min

Q: What is the angular speed $$\omega (in $$radians per second) of a wheel that makes 3 revolutions in 6 seconds?

$$\omega = \frac{3 \times 2\pi}{6} = \pi$$ radians/second

Q: A bicycle wheel has a diameter of 28 inches. If the bike is traveling at 15 miles per hour, what is the angular speed of the wheel in radians per hour? (Use $$v = r\omega$$ and $$r = 14$$ inches.)

$$\omega = \frac{15 \times 63360}{14}$$ radians/hour

Q: Which of the following is the correct formula for converting degrees to radians?

$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$

Q: A car is driving around a circular track of radius 100 meters at a constant speed. If it completes one revolution in 50 seconds, what is its linear speed?

$$v = \frac{2\pi \times 100}{50} = 4\pi m/s$$

Q: A Ferris wheel with a radius of 20 meters completes one revolution every 2 minutes. What is the linear speed of a seat on the edge of the wheel?

$$v = \frac{2\pi \times 20}{2} = 20\pi m/$$min

Question 1

Which of the following best describes an angle in standard position?

Accepted Answer

Its vertex is at the origin and its initial side is on the positive x-axis.

Question 2

If an angle is measured as $$-120^\circ$$, which direction is the rotation and in which quadrant does its terminal side lie?

Accepted Answer

Clockwise; Quadrant III

Question 3

Convert $$210^\circ to $$radians.

Accepted Answer

$$\frac{7\pi}{6}$$

Question 4

A circle has a radius of 5 meters. What is the length of the arc subtended by a central angle of $$2$$ radians?

Accepted Answer

$$10$$ meters

Question 5

Which formula gives the area $$A of a $$sector of a circle of radius $$r$$ formed by a central angle $$	heta (in $$radians)?

Accepted Answer

$$A = \frac{1}{2} r^2$ 	heta$$

Question 6

A water sprinkler sprays water over a distance of 10 meters while rotating through an angle of $$120^\circ. $$What is the area of the lawn that receives water? (Use $$A = \frac{1}{2} r^2$ 	heta$$ and convert angle to radians.)

Accepted Answer

$$\frac{1}{2} 	imes 10^2$ 	imes \frac{2\pi}{3} = 100\pi/3$$ square meters

Question 7

A child spins a rock at the end of a 2-foot rope at 180 revolutions per minute. What is the linear speed of the rock when released? (Use $$v = r\omega$$ and $$\omega in $$radians per minute.)

Accepted Answer

$$v = 2 	imes 180 	imes 2\pi = 720\pi$$ feet/min

Question 8

What is the angular speed $$\omega (in $$radians per second) of a wheel that makes 3 revolutions in 6 seconds?

Accepted Answer

$$\omega = \frac{3 	imes 2\pi}{6} = \pi$$ radians/second

Question 9

A bicycle wheel has a diameter of 28 inches. If the bike is traveling at 15 miles per hour, what is the angular speed of the wheel in radians per hour? (Use $$v = r\omega$$ and $$r = 14$$ inches.)

Accepted Answer

$$\omega = \frac{15 	imes 63360}{14}$$ radians/hour

Question 10

Which of the following is the correct formula for converting degrees to radians?

Accepted Answer

$$	ext{radians} = 	ext{degrees} 	imes \frac{\pi}{180}$$

Question 11

If a central angle of $$\frac{\pi}{3}$$ radians subtends an arc of length 8 cm on a circle, what is the radius of the circle?

Accepted Answer

$$r = \frac{8}{\frac{\pi}{3}} = \frac{24}{\pi} cm$$

Question 12

A car is driving around a circular track of radius 100 meters at a constant speed. If it completes one revolution in 50 seconds, what is its linear speed?

Accepted Answer

$$v = \frac{2\pi 	imes 100}{50} = 4\pi m/s$$

Question 13

Which of the following angles is a quadrantal angle?

Accepted Answer

$$270^\circ$$

Question 14

If an angle measures $$495^\circ$$, what is its coterminal angle between $$0^\circ$$ and $$360^\circ$$?

Accepted Answer

$$135^\circ$$

Question 15

A central angle of $$	heta$$ radians in a circle of radius $$r$$ subtends an arc of length $$s. $$Which equation correctly relates these quantities?

Accepted Answer

$$s = r	heta$$

Question 16

A Ferris wheel with a radius of 20 meters completes one revolution every 2 minutes. What is the linear speed of a seat on the edge of the wheel?

Accepted Answer

$$v = \frac{2\pi 	imes 20}{2} = 20\pi m/$$min

Angles, Radians, Arc Length, and Circular Motion – Precalculus Study Notes

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