Q: A wheel with a 28-inch radius is marked at two points on the rim. The distance between the marks along the wheel is found to be 16 inches. What is the angle (in degrees) between the radii to the two marks? (Round to the nearest tenth of a degree.)

$$\theta = \dfrac{16}{28} \times 180 / \pi \approx 32.7^\circ$$

Q: Given $$\sin \theta = \dfrac{40}{41}$$ and $$\cos \theta = \dfrac{9}{41}$$, find $$\sin(-\theta)$$ and $$\cos(-\theta).$$

$$\sin(-\theta) = -\dfrac{40}{41}$$, $$\cos(-\theta) = \dfrac{9}{41}$$

Q: Given $$\sin \theta = \dfrac{1}{2}$$ and $$0 \u003c \theta \u003c \dfrac{\pi}{2}$$, use the Pythagorean identity to find $$\cos \theta.$$

$$\cos \theta = \dfrac{\sqrt{3}}{2}$$

Q: Given the right triangle with sides $$7 ($$adjacent), $$24 ($$opposite), and $$25 ($$hypotenuse), find $$\sin A$$ and $$\tan A.$$

$$\sin A = \dfrac{24}{25}$$, $$\tan A = \dfrac{24}{7}$$

Q: Which of the following is the reciprocal identity for $$\sec \theta$$?

$$\sec \theta = \dfrac{1}{\cos \theta}$$

Q: Which of the following is the correct Pythagorean identity?

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

Question 1

A wheel with a 28-inch radius is marked at two points on the rim. The distance between the marks along the wheel is found to be 16 inches. What is the angle (in degrees) between the radii to the two marks? (Round to the nearest tenth of a degree.)

Accepted Answer

$$	heta = \dfrac{16}{28} 	imes 180 / \pi \approx 32.7^\circ$$

Question 2

Find the amplitude and period of the function $$y = 3 \cos 5x.$$

Accepted Answer

Amplitude: $$3$$, Period: $$\dfrac{2\pi}{5}$$

Question 3

Given $$\sin 	heta = \dfrac{40}{41}$$ and $$\cos 	heta = \dfrac{9}{41}$$, find $$\sin(-	heta)$$ and $$\cos(-	heta).$$

Accepted Answer

$$\sin(-	heta) = -\dfrac{40}{41}$$, $$\cos(-	heta) = \dfrac{9}{41}$$

Question 4

Which of the following is NOT part of the range of the tangent inverse function $$	an$$^{-1}$$x$$?

Accepted Answer

$$\dfrac{\pi}{2}$$

Question 5

An object moves in simple harmonic motion described by $$d = 2 \cos 3\pi t$$ meters, where $$t is in $$seconds. What is the frequency of the motion?

Accepted Answer

$$\dfrac{3}{2}$$ oscillations/second

Question 6

Find the exact value of $$\sin 	heta if $$the point $$(21, 28) is on $$the terminal side of angle $$	heta.$$

Accepted Answer

$$\dfrac{28}{35}$$

Question 7

Which of the following is the correct formula for arc length $$s in $$terms of radius $$r$$ and angle $$	heta in $$radians?

Accepted Answer

$$s = r	heta$$

Question 8

Given $$\sin 	heta = \dfrac{1}{2}$$ and $$0 \u003c 	heta \u003c \dfrac{\pi}{2}$$, use the Pythagorean identity to find $$\cos 	heta.$$

Accepted Answer

$$\cos 	heta = \dfrac{\sqrt{3}}{2}$$

Question 9

Find the period and phase shift for the graph of $$y = \cos(7x + 4).$$

Accepted Answer

Period: $$\dfrac{2\pi}{7}$$, Phase shift: $$-\dfrac{4}{7}$$

Question 10

If $$y = \sin x$$ and $$y = \cos x$$, which of the following statements about their domains and ranges is true?

Accepted Answer

Both have domain $$(-\infty, \infty)$$ and range $$[-1, 1]$$

Question 11

Given the right triangle with sides $$7 ($$adjacent), $$24 ($$opposite), and $$25 ($$hypotenuse), find $$\sin A$$ and $$	an A.$$

Accepted Answer

$$\sin A = \dfrac{24}{25}$$, $$	an A = \dfrac{24}{7}$$

Question 12

Which of the following is the reciprocal identity for $$\sec 	heta$$?

Accepted Answer

$$\sec 	heta = \dfrac{1}{\cos 	heta}$$

Question 13

A minute hand of a clock is 8 inches long. How far does the tip of the minute hand move in 25 minutes? (Round to two decimal places.)

Accepted Answer

$$33.49$$ inches

Question 14

Which of the following is the correct Pythagorean identity?

Accepted Answer

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

Question 15

Write an equation for a simple harmonic motion with a frequency of $$4$$ vibrations per second and maximum displacement $$5$$ meters.

Accepted Answer

$$d = 5 \sin(8\pi t)$$

Question 16

Which of the following is the correct domain and range for $$y = 	an x$$?

Accepted Answer

Domain: $$x 
eq \dfrac{\pi}{2} + k\pi$$, Range: $$(-\infty, \infty)$$

Question 17

If $$y = -2 \cos\left(\dfrac{x}{2}\right)$$, what is the period of the function?

Accepted Answer

$$4\pi$$

Question 18

Given $$	an 	heta = \dfrac{3}{4}$$ and $$	heta is in $$the first quadrant, what is $$\sin 	heta$$?

Accepted Answer

$$\dfrac{3}{5}$$

Question 19

Which of the following is the correct cofunction identity?

Accepted Answer

$$\sin(\dfrac{\pi}{2} - x) = \cos x$$

Precalculus Study Guide: Trigonometric Functions, Identities, and Applications

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