Q: A circle has a radius of 12 cm. What is the length of the arc intercepted by a central angle of $$60^\\circ$$?

$$s = 12 \\left(\\frac{\\pi}{3}\\right) = 4\\pi cm$$

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

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

Q: A belt runs a pulley of radius 8 cm at 60 revolutions per minute. What is the angular speed of the pulley in radians per second?

$$\\frac{16\\pi}{3}$$ radians/sec

Q: Convert $$2.5$$ radians to degrees.

$$2.5 \\times \\frac{180^\\circ}{\\pi} \\approx 143.24^\\circ$$

Q: A point on a circle of radius 5 meters moves with an angular speed of $$2$$ radians per second. What is its linear speed?

$$v = r\\omega = 5 \\times 2 = 10 m/s$$

Q: The latitude of City A is $$52^\\circ N $$and City B is $$40^\\circ N. If $$the radius of Earth is $$6371 km$$, what is the north-south distance between the two cities?

$$s = 6371 \\times \\left(\\frac{12\\pi}{180}\\right) \\approx 1335 km$$

Q: Which of the following is true about complementary angles?

Their measures add up to $$90^\\circ.$$

Q: A sector of a circle has a radius of 10 m and a central angle of $$2$$ radians. What is the area of the sector?

$$A = \\frac{1}{2} \\times 10^2$ \\times 2 = 100 m$^2$$

Q: A central angle of $$x$$ radians in a circle of radius $$r$$ intercepts an arc of length $$s. If$$ $$s = 15 cm $$and $$r = 5 cm$$, what is $$x$$?

$$x = \\frac{15}{5} = 3$$ radians

Question 1

Which of the following best describes a quadrantal angle?

Accepted Answer

An angle whose terminal side lies along the x- or y-axis in standard position.

Question 2

Convert $$225^\circ to $$radians.

Accepted Answer

$$\frac{5\pi}{4}$$ radians

Question 3

A circle has a radius of 12 cm. What is the length of the arc intercepted by a central angle of $$60^\circ$$?

Accepted Answer

$$s = 12 \left(\frac{\pi}{3}\right) = 4\pi cm$$

Question 4

If two angles are supplementary and one angle measures $$x$$, what is the measure of the other angle?

Accepted Answer

$$180^\circ - x$$

Question 5

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

Accepted Answer

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

Question 6

A belt runs a pulley of radius 8 cm at 60 revolutions per minute. What is the angular speed of the pulley in radians per second?

Accepted Answer

$$\frac{16\pi}{3}$$ radians/sec

Question 7

Which of the following angles is coterminal with $$-45^\circ$$?

Accepted Answer

$$315^\circ$$

Question 8

Convert $$2.5$$ radians to degrees.

Accepted Answer

$$2.5 \times \frac{180^\circ}{\pi} \approx 143.24^\circ$$

Question 9

A point on a circle of radius 5 meters moves with an angular speed of $$2$$ radians per second. What is its linear speed?

Accepted Answer

$$v = r\omega = 5 \times 2 = 10 m/s$$

Question 10

The latitude of City A is $$52^\circ N $$and City B is $$40^\circ N. If $$the radius of Earth is $$6371 km$$, what is the north-south distance between the two cities?

Accepted Answer

$$s = 6371 \times \left(\frac{12\pi}{180}\right) \approx 1335 km$$

Question 11

Which of the following is true about complementary angles?

Accepted Answer

Their measures add up to $$90^\circ.$$

Question 12

Convert $$120^\circ 30' 0'' to $$decimal degrees.

Accepted Answer

$$120.5^\circ$$

Question 13

A sector of a circle has a radius of 10 m and a central angle of $$2$$ radians. What is the area of the sector?

Accepted Answer

$$A = \frac{1}{2} \times 10^2$ \times 2 = 100 m$^2$$

Question 14

Which of the following is the correct relationship between degrees and radians?

Accepted Answer

$$180^\circ = \pi$$ radians

Question 15

If a point moves along a circle of radius $$r$$ and subtends an angle $$\theta (in $$radians) at the center, what is the formula for the arc length $$s$$?

Accepted Answer

$$s = r\theta$$

Question 16

A central angle of $$x$$ radians in a circle of radius $$r$$ intercepts an arc of length $$s. If$$ $$s = 15 cm $$and $$r = 5 cm$$, what is $$x$$?

Accepted Answer

$$x = \frac{15}{5} = 3$$ radians

Angles, Arc, and Their Measures – Chapter 9.1 Study Notes (Precalculus: Trigonometry Functions and Applications)

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