Q: Which of the following best explains why the coordinates on the unit circle correspond to $$(\cos \theta, \sin \theta)$$?

Because for any angle $$\theta$$, the $$x-$$coordinate is the adjacent side (cosine) and the $$y-$$coordinate is the opposite side (sine) of the right triangle formed with the radius $$1.$$

Question 1

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

Accepted Answer

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

Question 2

A circle has a radius of $$6$$ units. What is the arc length subtended by a central angle of $$\frac{\pi}{2}$$ radians?

Accepted Answer

$$3\pi$$

Question 3

If $$\sin 	heta = -\frac{3}{5}$$ and $$	heta is in $$Quadrant IV, what is $$\cos 	heta$$?

Accepted Answer

$$\frac{4}{5}$$

Question 4

Which of the following best describes what it means geometrically for an angle to measure $$1$$ radian?

Accepted Answer

It subtends an arc equal in length to the radius of the circle.

Question 5

A Ferris wheel has a radius of $$20 ft. If a $$rider rotates $$\frac{3\pi}{4}$$ radians, how far does the rider travel along the arc?

Accepted Answer

$$15\pi ft$$

Question 6

Convert $$\frac{5\pi}{6}$$ radians to degrees.

Accepted Answer

$$150^\circ$$

Question 7

If $$\cos 	heta = -\frac{2}{3}$$ and $$	heta is in $$Quadrant III, what is $$\sin 	heta$$?

Accepted Answer

$$-\frac{\sqrt{5}}{3}$$

Question 8

Which quadrants have positive sine values?

Accepted Answer

Quadrants I and II

Question 9

A triangle has sides: Opposite $$= 5$$, Adjacent $$= 12. $$What is the hypotenuse?

Accepted Answer

$$13$$

Question 10

If the radius of a circle doubles, what happens to the arc length for a fixed central angle?

Accepted Answer

The arc length doubles.

Question 11

Which of the following is the correct amplitude, midline, and period for $$f(x) = 4 \sin x$$?

Accepted Answer

Amplitude: $$4$$, Midline: $$0$$, Period: $$2\pi$$

Question 12

Given a point $$(3, 4) on $$the terminal side of an angle $$	heta$$, what is $$\sin 	heta$$?

Accepted Answer

$$\frac{4}{5}$$

Question 13

Which trigonometric functions are positive in Quadrant II?

Accepted Answer

Sine and cosecant

Question 14

What are the coordinates on the unit circle for $$60^\circ$$?

Accepted Answer

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

Question 15

If $$\sin 	heta = 4/5$$ and $$	heta is $$acute, what is $$\cos 	heta$$?

Accepted Answer

$$3/5$$

Question 16

Which of the following best explains why the coordinates on the unit circle correspond to $$(\cos 	heta, \sin 	heta)$$?

Accepted Answer

Because for any angle $$	heta$$, the $$x-$$coordinate is the adjacent side (cosine) and the $$y-$$coordinate is the opposite side (sine) of the right triangle formed with the radius $$1.$$

Question 17

What is the period of $$y = \sin \left(\frac{\pi}{2} x\right)$$?

Accepted Answer

$$4$$

Question 18

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

Accepted Answer

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

Question 19

A circle has radius $$5$$ and an arc length of $$10$$ units. What is the angle in radians subtended by the arc?

Accepted Answer

$$2$$

Question 20

If $$f(x) = 4 \sin \left(\frac{\pi}{2} x\right) + 7.5$$, what is the midline of the function?

Accepted Answer

$$7.5$$

Precalculus Study Guide: Angles, Radians, Unit Circle, and Trigonometric Concepts

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