Q: The latitude of City A is $$50^{\circ} N $$and City B is $$40^{\circ} N. If $$the radius of Earth is 6400 km, what is the north-south distance between the two cities?

$$6400 \times \frac{10\pi}{180} km$$

Q: Which of the following is the correct formula for the area $$A of a $$sector of a circle with radius $$r$$ and central angle $$\theta (in $$radians)?

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

Q: A rope is wound around a drum of radius $$0.8725 ft $$through an angle of $$39.72^{\circ}. $$How much rope is wound? (Express your answer in terms of $$\pi.)$$

$$0.8725 \times \frac{39.72\pi}{180} ft$$

Q: If the smaller gear (radius 2.5 cm) rotates through $$225^{\circ}$$, how many degrees does the larger gear (radius 4.8 cm) rotate?

$$225^{\circ}$$ \times \frac{2.5}{4.8}$$

Q: A sector-shaped field has a radius of 324 m and a central angle of $$15^{\circ}. $$What is the area of the field?

$$\frac{1}{2} \times 324^2$ \times \frac{15\pi}{180}$$

Question 1

Which of the following best describes a radian?

Accepted Answer

An angle that intercepts an arc equal in length to the radius of the circle.

Question 2

Convert $$135^{\circ} to $$radians.

Accepted Answer

$$\frac{3\pi}{4}$$

Question 3

A circle has a radius of 10 cm. What is the length of the arc intercepted by a central angle of $$\frac{\pi}{3}$$ radians?

Accepted Answer

$$\frac{10\pi}{3} cm$$

Question 4

The latitude of City A is $$50^{\circ} N $$and City B is $$40^{\circ} N. If $$the radius of Earth is 6400 km, what is the north-south distance between the two cities?

Accepted Answer

$$6400 	imes \frac{10\pi}{180} km$$

Question 5

Find the area of a sector with radius 8 m and central angle $$60^{\circ}.$$

Accepted Answer

$$\frac{16\pi}{3} m$^2$$

Question 6

Which of the following is the correct formula for the area $$A of a $$sector of a circle with radius $$r$$ and central angle $$	heta (in $$radians)?

Accepted Answer

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

Question 7

If a point moves along a circle of radius $$r at a $$constant angular speed $$\omega$$, what is the formula for its linear speed $$v$$?

Accepted Answer

$$v = r \omega$$

Question 8

A pulley with radius 6 cm rotates at 80 revolutions per minute. What is its angular speed in radians per second?

Accepted Answer

$$\frac{80 	imes 2\pi}{60}$$

Question 9

On the unit circle, what are the coordinates of the point corresponding to an angle of $$\frac{\pi}{6}$$?

Accepted Answer

$$\left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right)$$

Question 10

If $$\sin s = 0.5$$ and $$s is in $$the interval $$[0, \pi]$$, what is $$s$$?

Accepted Answer

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

Question 11

A satellite orbits Earth at a height of 1600 km above the surface. If the radius of Earth is 6400 km, what is the radius of the satellite's orbit?

Accepted Answer

$$8000 km$$

Question 12

Which of the following statements about radians is TRUE?

Accepted Answer

Radians are a pure number, not a unit like degrees.

Question 13

If the linear speed of a point on a circle is 5 m/s and the radius is 2 m, what is the angular speed?

Accepted Answer

$$2.5$$ rad/s

Question 14

Which of the following is the correct relationship between the arc length $$s$$, radius $$r$$, and central angle $$	heta (in $$radians)?

Accepted Answer

$$s = r 	heta$$

Question 15

A rope is wound around a drum of radius $$0.8725 ft $$through an angle of $$39.72^{\circ}. $$How much rope is wound? (Express your answer in terms of $$\pi.)$$

Accepted Answer

$$0.8725 	imes \frac{39.72\pi}{180} ft$$

Question 16

If the smaller gear (radius 2.5 cm) rotates through $$225^{\circ}$$, how many degrees does the larger gear (radius 4.8 cm) rotate?

Accepted Answer

$$225^{\circ}$$ 	imes \frac{2.5}{4.8}$$

Question 17

Which of the following is NOT a property of the unit circle?

Accepted Answer

Its radius is always 2 units.

Question 18

If $$\cos s = 0.9685$$ and $$s is in$$ $$[0, \pi]$$, what is the approximate value of $$s (in $$radians)?

Accepted Answer

$$0.25$$

Question 19

A sector-shaped field has a radius of 324 m and a central angle of $$15^{\circ}. $$What is the area of the field?

Accepted Answer

$$\frac{1}{2} 	imes 324^2$ 	imes \frac{15\pi}{180}$$

Chapter 3: Radian Measure and the Unit Circle – Precalculus Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards