$$\theta \approx 44.4^\circ, 135.6^\circ$$

Question 1

Given $$\sin 	heta = \frac{4}{5}$$ and $$	heta is in $$Quadrant II, what is $$\cos 	heta$$?

Accepted Answer

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

Question 2

If $$\cos 	heta = \frac{8}{10}$$ and $$	heta is in $$Quadrant IV, what is $$	an 	heta$$?

Accepted Answer

$$-\frac{3}{4}$$

Question 3

Solve for $$x in $$the interval $$[0, 2\pi)$$: $$(	an x + 1)(2 \sin x - 1) = 0.$$

Accepted Answer

$$x = \frac{3\pi}{4}, \frac{7\pi}{4}, \frac{\pi}{6}, \frac{5\pi}{6}$$

Question 4

What is $$\cos\left(\frac{\pi}{7}\right)$$?

Accepted Answer

$$\cos\left(\frac{\pi}{7}\right)$$

Question 5

Which of the following is a solution to $$\cos x = \sqrt{1 - \sin^2$ x}$$?

Accepted Answer

$$x = 0$$

Question 6

Evaluate $$\cos$$^{-1}$$\left(\frac{1}{2}\right) in $$radians.

Accepted Answer

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

Question 7

Solve $$2 	an\left(x - \frac{\pi}{2}ight) \sqrt{3} = 0$$ for $$x in$$ $$[0, 2\pi).$$

Accepted Answer

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

Question 8

If $$	an u = \sec u - 1$$, which of the following is equivalent to $$	an u$$?

Accepted Answer

$$\frac{\cos u}{1 - \sin u}$$

Question 9

Given $$y = \sin$$^{-1}$$\left(\sin\frac{7\pi}{11}\right)$$, what is the exact value of $$y$$?

Accepted Answer

$$y = \frac{7\pi}{11}$$

Question 10

All solutions to $$\sin x = -\frac{1}{2}$$ are of the form:

Accepted Answer

$$x = 7\pi/6 + 2\pi n$$ and $$x = 11\pi/6 + 2\pi n$$, where $$n is $$any integer

Question 11

Evaluate $$y = \cos\left(\sin$$^{-1}$$\frac{1}{3}\right).$$

Accepted Answer

$$y = \frac{2\sqrt{2}}{3}$$

Question 12

Given $$\cos 	heta = \frac{8}{10}$$ and $$	heta is in $$Quadrant IV, find $$\sin 2	heta.$$

Accepted Answer

$$-\frac{12}{25}$$

Question 13

Find all solutions to $$\sin 	heta = 0.7 in $$the interval $$[0, 360^\circ].$$

Accepted Answer

$$	heta \approx 44.4^\circ, 135.6^\circ$$

Question 14

Use a cosine sum or difference identity to find the exact value of $$\cos\left(\frac{5\pi}{12}\right).$$

Accepted Answer

$$\frac{\sqrt{6} - \sqrt{2}}{4}$$

Question 15

Evaluate $$\sin(45^\circ + 180^\circ).$$

Accepted Answer

$$-\frac{\sqrt{2}}{2}$$

Question 16

Find all solutions to $$\cos\left(x + \frac{\pi}{2}\right) = \cos x in $$the interval $$[0, 2\pi).$$

Accepted Answer

$$x = 0, \pi$$

Question 17

Express the identity $$\sin\left(\frac{\pi}{2} - 	hetaight) \cos 	heta in $$terms of $$\sin 	heta$$ and $$\cos 	heta.$$

Accepted Answer

$$\cos^2$ 	heta$$

Question 18

Find all values of $$x in$$ $$[0, 2\pi)$$ such that $$\sin(x) = -1.$$

Accepted Answer

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

Question 19

Given $$\sin(a) = \frac{3}{5}$$ with $$a in $$Quadrant II, and $$\cos(b) = \frac{2}{3}$$ with $$b in $$Quadrant IV, find $$\sin(a-b).$$

Accepted Answer

$$\frac{13}{15}$$

Question 20

Simplify $$\cos x \sin x \sqrt{\cos^2$ x + \sec^2$ x}.$$

Accepted Answer

$$\sin x \sec x$$

Precalculus Trigonometry: Identities, Equations, and Exact Values

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