Q: Which of the following is the reciprocal identity for $$\sin(\theta)$$?

$$\csc(\theta) = \frac{1}{\sin(\theta)}$$

Q: Which of the following is always true for any angle $$\theta on $$the unit circle?

$$\sin^2$(\theta) + \cos^2$(\theta) = 1$$

Question 1

Which of the following best describes what is being measured when we measure an angle in a circle?

Accepted Answer

The amount of full rotation from an initial side to a terminal side, independent of the circle's size

Question 2

If an angle measures $$1$$ radian, what does this mean in terms of arc length and radius?

Accepted Answer

The arc length is equal to the radius of the circle

Question 3

Convert $$2.7$$ radians to degrees. (Round to the nearest tenth)

Accepted Answer

$$154.7^{\circ}$$

Question 4

Convert $$90^{\circ} to $$radians.

Accepted Answer

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

Question 5

Given a circle with radius $$r = 2$$ inches and a central angle $$	heta = 0.45$$ radians, what is the arc length $$s$$ cut off by the angle?

Accepted Answer

$$0.9$$ inches

Question 6

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

Accepted Answer

$$s = r \cdot 	heta$$

Question 7

A central angle of $$4$$ radians cuts off an arc in a circle of radius $$3 cm. $$What is the arc length?

Accepted Answer

$$12 cm$$

Question 8

What is the degree measure of $$\pi$$ radians?

Accepted Answer

$$180^{\circ}$$

Question 9

If the radius of a circle is doubled, how does the arc length corresponding to a fixed central angle (in radians) change?

Accepted Answer

It doubles

Question 10

A sector of a circle has a central angle of $$60^{\circ}$$ and a radius of $$5 cm. $$What is the area of the sector? (Use $$A = \frac{1}{2} r^2$ 	heta$$ with $$	heta in $$radians)

Accepted Answer

$$\frac{25\pi}{6} cm$^2$$

Question 11

Which of the following is the correct conversion factor to change degrees to radians?

Accepted Answer

$$\frac{\pi}{180$$^{\circ}$$}$$

Question 12

On the unit circle, what are the coordinates of the point corresponding to an angle of $$0$$ radians?

Accepted Answer

$$(1, 0)$$

Question 13

If $$\sin(	heta) = \frac{1}{2}$$ and $$	heta is in $$the first quadrant, what is $$	heta in $$radians?

Accepted Answer

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

Question 14

Which of the following is the reciprocal identity for $$\sin(	heta)$$?

Accepted Answer

$$\csc(	heta) = \frac{1}{\sin(	heta)}$$

Question 15

A Ferris wheel has a radius of $$10$$ meters. If a passenger travels through a central angle of $$\frac{\pi}{3}$$ radians, what distance along the wheel's edge have they traveled?

Accepted Answer

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

Question 16

Which of the following is always true for any angle $$	heta on $$the unit circle?

Accepted Answer

$$\sin^2$(	heta) + \cos^2$(	heta) = 1$$

Question 17

If $$	an(	heta) = 1$$ and $$	heta is in $$the first quadrant, what is $$	heta$$?

Accepted Answer

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

Question 18

A central angle of $$2$$ radians in a circle of radius $$5$$ meters cuts off an arc. What is the arc length?

Accepted Answer

$$10$$ meters

Question 19

Which of the following statements about radians is correct?

Accepted Answer

A radian is the angle that cuts off an arc equal in length to the radius

Question 20

If $$\cos(	heta) = 0$$ and $$0 \leq 	heta \u003c 2\pi$$, what are the possible values of $$	heta$$?

Accepted Answer

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

Unit 4: Circle Trigonometry – Angles, Radians, and Arc Length

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