Q: A central angle in a circle of radius 10 cm intercepts an arc of length 25 cm. What is the radian measure of the angle?

$$\theta = \frac{25}{10} = 2.5$$ radians

Q: If a point moves along a circle of radius $$8$$ meters and sweeps out a central angle of $$\frac{\pi}{3}$$ radians, what is the length of the arc it travels?

$$s = r\theta = 8 \times \frac{\pi}{3} = \frac{8\pi}{3}$$ meters

Q: Convert $$\frac{7\pi}{6}$$ radians to degrees.

$$\frac{7\pi}{6} \times \frac{180$$^{\circ}$$}{\pi} = 210$$^{\circ}$$

Q: A 45-rpm record has a radius of $$1.5$$ inches. What is the linear speed at the edge of the record in inches per minute? (Use $$\omega = 45 \times 2\pi$$ radians/minute)

$$v = r\omega = 1.5 \times 90\pi = 135\pi$$ inches/minute

Q: A car tire with a radius of $$0.3$$ meters rotates at an angular speed of $$10$$ radians per second. What is the linear speed of a point on the edge of the tire?

$$v = r\omega = 0.3 \times 10 = 3 m/s$$

Question 1

Which of the following best describes a quadrantal angle?

Accepted Answer

An angle whose terminal side lies on the x-axis or y-axis.

Question 2

A central angle in a circle of radius 10 cm intercepts an arc of length 25 cm. What is the radian measure of the angle?

Accepted Answer

$$	heta = \frac{25}{10} = 2.5$$ radians

Question 3

Convert $$120^{\circ} to $$radians.

Accepted Answer

$$\frac{2\pi}{3}$$ radians

Question 4

Which of the following angles is coterminal with $$-135^{\circ}$$ and is positive and less than $$360^{\circ}$$?

Accepted Answer

$$225^{\circ}$$

Question 5

A wheel rotates through $$5\pi$$ radians. How many full revolutions has it completed?

Accepted Answer

$$\frac{5\pi}{2\pi} = 2.5$$ revolutions

Question 6

If a point moves along a circle of radius $$8$$ meters and sweeps out a central angle of $$\frac{\pi}{3}$$ radians, what is the length of the arc it travels?

Accepted Answer

$$s = r	heta = 8 	imes \frac{\pi}{3} = \frac{8\pi}{3}$$ meters

Question 7

Which of the following statements about positive and negative angles is correct?

Accepted Answer

Positive angles are generated by counterclockwise rotation, and negative angles by clockwise rotation.

Question 8

Convert $$\frac{7\pi}{6}$$ radians to degrees.

Accepted Answer

$$\frac{7\pi}{6} 	imes \frac{180$$^{\circ}$$}{\pi} = 210$$^{\circ}$$

Question 9

A 45-rpm record has a radius of $$1.5$$ inches. What is the linear speed at the edge of the record in inches per minute? (Use $$\omega = 45 	imes 2\pi$$ radians/minute)

Accepted Answer

$$v = r\omega = 1.5 	imes 90\pi = 135\pi$$ inches/minute

Question 10

Which formula correctly relates the arc length $$s$$, radius $$r$$, and central angle $$	heta (in $$radians) of a circle?

Accepted Answer

$$s = r	heta$$

Question 11

If an angle in standard position has its terminal side along the negative y-axis, what is its degree measure?

Accepted Answer

$$270^{\circ}$$

Question 12

Which of the following is a coterminal angle with $$\frac{13\pi}{5}$$ radians that is positive and less than $$2\pi$$?

Accepted Answer

$$\frac{13\pi}{5} - 2\pi = \frac{3\pi}{5}$$ radians

Question 13

A car tire with a radius of $$0.3$$ meters rotates at an angular speed of $$10$$ radians per second. What is the linear speed of a point on the edge of the tire?

Accepted Answer

$$v = r\omega = 0.3 	imes 10 = 3 m/s$$

Question 14

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

Accepted Answer

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

Question 15

Convert $$-300^{\circ} to $$radians.

Accepted Answer

$$-\frac{5\pi}{3}$$ radians

Question 16

A central angle of $$\frac{\pi}{4}$$ radians in a circle of radius $$6$$ inches intercepts an arc. What is the length of the arc, rounded to two decimal places?

Accepted Answer

$$s = 6 	imes \frac{\pi}{4} = \frac{3\pi}{2} \approx 4.71$$ inches

Question 17

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

Accepted Answer

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

Angles and Radian Measure: Foundations of Trigonometry

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