Q: Solve for all exact solutions: $$\sin \theta \sec \theta - \sin \theta = 0.$$

$$\{90^\circ n, \text{where } n \text{ is any integer}\}$$

Question 1

Given $$	an \alpha = -\frac{1}{4}$$ and $$\alpha is in $$quadrant IV, what is $$\cot \alpha$$?

Accepted Answer

$$4$$

Question 2

Simplify $$\sec^2$(-	heta) - \cos^2$(-	heta) - 	an^2$(-	heta) in $$terms of sine and cosine only.

Accepted Answer

$$1$$

Question 3

Verify the identity: $$\frac{1 + \cos x}{1 - \cos x} \cdot \frac{1 - \cos x}{1 + \cos x} = 4 \cot x \csc x. $$Which of the following steps is correct in the verification process?

Accepted Answer

Rewrite $$1 - \cos^2$ x as$$ $$\sin^2$ x$$

Question 4

Find the exact value: $$\cos\left(\frac{7\pi}{12}\right).$$

Accepted Answer

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

Question 5

Given $$\cos s = -\frac{3}{5}$$ and $$\sin t = -\frac{1}{5}$$ with $$s$$ and $$t in $$quadrant III, find $$\cos(s + t). $$

Accepted Answer

$$\frac{11}{25}$$

Question 6

Given $$	an x = 2$$ and $$\cos x \u003e 0$$, find $$\cos 2x.$$

Accepted Answer

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

Question 7

Use a half-angle identity to find the exact value of $$\cos 165^\circ.$$

Accepted Answer

$$-\frac{\sqrt{2 + \sqrt{3}}}{2}$$

Question 8

Given $$\cos \alpha = \frac{1}{3}$$ and $$0 \u003c \alpha \u003c \frac{\pi}{2}$$, determine the exact value of $$\cos \frac{\alpha}{2}.$$

Accepted Answer

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

Question 9

Given $$\sin 	heta = -\frac{5}{13}$$ and $$270^\circ \u003c 	heta \u003c 360^\circ$$, find $$	an \frac{	heta}{2}.$$

Accepted Answer

$$-\frac{12}{5}$$

Question 10

What is the domain of the function $$y = \arcsin x$$?

Accepted Answer

$$[-1, 1]$$

Question 11

Find the exact value: $$y = \sin$$^{-1}$$\left(\frac{\sqrt{3}}{2}\right).$$

Accepted Answer

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

Question 12

Find the exact value: $$y = \arcsin\left(-\frac{\sqrt{3}}{2}\right).$$

Accepted Answer

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

Question 13

Find the approximate value, in degrees, of $$\csc$$^{-1}$$(3.33). ($$Round to two decimal places.)

Accepted Answer

$$17.54^\circ$$

Question 14

Give the exact value: $$\cos\left(	an$$^{-1}$$ 4ight).$$

Accepted Answer

$$\frac{1}{\sqrt{17}}$$

Question 15

Give the exact value: $$\cos\left(2 \arctan\left(\frac{5}{12}\right)\right).$$

Accepted Answer

$$\frac{119}{169}$$

Question 16

Write $$\sin(\arccos u) as an $$algebraic expression in $$u$$ for $$u \u003e 0.$$

Accepted Answer

$$\sqrt{1 - u^2$}$$

Question 17

Solve for $$x$$ over $$[0, 2\pi)$$: $$-2\cos^2$ x = -\cos x - 1.$$

Accepted Answer

$$\left\{\frac{2\pi}{3}, \frac{4\pi}{3}\right\}$$

Question 18

Solve for $$	heta$$ over $$[0^\circ, 360^\circ)$$: $$2\cos 	heta + 1 = \sec 	heta.$$

Accepted Answer

$$60^\circ, 300^\circ$$

Question 19

Solve for all exact solutions: $$\sin 	heta \sec 	heta - \sin 	heta = 0.$$

Accepted Answer

$$\{90^\circ n, 	ext{where } n 	ext{ is any integer}\}$$

Question 20

Solve for $$x$$ over $$[0, 2\pi)$$: $$7	an 3x = 7.$$

Accepted Answer

$$\left\{\frac{\pi}{12}, \frac{5\pi}{12}, \frac{9\pi}{12}\right\}$$

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