Q: Which of the following is the correct domain and range for the inverse sine function $$y = \arcsin x$$?

$$\text{Domain: } [-1, 1],\ \text{Range: } \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$

Question 1

Which of the following is the correct domain and range for the inverse sine function $$y = \arcsin x$$?

Accepted Answer

$$	ext{Domain: } [-1, 1],\ 	ext{Range: } \left[-\frac{\pi}{2}, \frac{\pi}{2}ight]$$

Question 2

Evaluate $$\sin(\arccos(\frac{1}{2})).$$

Accepted Answer

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

Question 3

Which of the following identities is a Pythagorean identity?

Accepted Answer

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

Question 4

Solve for $$x in$$ $$[0, 2\pi)$$: $$2\sin^2$ x - 3\sin x + 1 = 0.$$

Accepted Answer

$$x = \frac{\pi}{6},\ \frac{5\pi}{6},\ \frac{3\pi}{2}$$

Question 5

Which of the following is the correct expression for $$\cos(\alpha + \beta)$$?

Accepted Answer

$$\cos \alpha \cos \beta - \sin \alpha \sin \beta$$

Question 6

If $$	an(\arcsin(\frac{3}{5})) = x$$, what is the value of $$x$$?

Accepted Answer

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

Question 7

Which of the following is the correct double angle identity for $$\sin 2x$$?

Accepted Answer

$$2\sin x \cos x$$

Question 8

Evaluate $$\arccos(\cos(\frac{5\pi}{6})).$$

Accepted Answer

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

Question 9

Which of the following is a valid step when verifying a trigonometric identity?

Accepted Answer

Convert all terms to sines and cosines.

Question 10

Given $$\sin u = \frac{12}{13}$$ and $$0 \u003c u \u003c \frac{\pi}{2}$$, $$\cos v = -\frac{3}{5}$$ and $$\frac{\pi}{2} \u003c v \u003c \pi$$, find $$\cos(u + v).$$

Accepted Answer

$$-\frac{56}{65}$$

Question 11

Which of the following is the correct reciprocal identity?

Accepted Answer

$$\csc x = \frac{1}{\sin x}$$

Question 12

Solve for $$x in$$ $$[0, 2\pi)$$: $$2\cos x + \sin 2x = 0.$$

Accepted Answer

$$x = \frac{2\pi}{3},\ \frac{4\pi}{3},\ \frac{3\pi}{2}$$

Question 13

Which of the following is the correct sum identity for $$\sin(a + b)$$?

Accepted Answer

$$\sin a \cos b + \cos a \sin b$$

Question 14

Evaluate $$	an(\arccos(\frac{4}{5})).$$

Accepted Answer

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

Question 15

Which of the following is the correct even-odd identity?

Accepted Answer

$$\sin(-x) = -\sin x$$

Question 16

If $$f(x) = 	an(\arccos(\frac{x}{2}))$$, what is $$f(1)$$?

Accepted Answer

$$\frac{\sqrt{3}}{1}$$

Question 17

Which of the following is the correct composition property for inverse trigonometric functions?

Accepted Answer

$$\sin(\arcsin x) = x$$ for $$-1 \leq x \leq 1$$

Analytic Trigonometry: Inverse Functions, Fundamental Identities, and Trigonometric Equations

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