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

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

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

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

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 $0 < u < \frac{\pi}{2}$, $\cos v = -\frac{3}{5}$ and $\frac{\pi}{2} < v < \pi$, 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

Study Guide - Practice Questions