Q: Which of the following is the correct double-angle identity for $\tan(2x)$?

$\tan(2x) = \frac{2 \tan x}{1 - \tan^2 x}$

Q: Which of the following is the correct half-angle identity for $\sin\left(\frac{\theta}{2}\right)$?

$\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$

Question 1

Which of the following is the correct period for the function $y = 4 \sin\left(\frac{1}{3}x - \frac{\pi}{2}\right)$?

Accepted Answer

$3\pi$

Question 2

Given $\cos 	heta = \frac{5}{13}$ and $\cos \phi = \frac{3}{5}$, with $	heta, \phi \in [0, \frac{\pi}{2})$, what is $\cos(	heta + \phi)$?

Accepted Answer

$\frac{56}{65}$

Question 3

Which trigonometric function has vertical asymptotes at $x = \frac{\pi}{2} + n\pi$, where $n \in \mathbb{Z}$?

Accepted Answer

$	an(x)$

Question 4

Simplify the expression $24 \cos^3 x \sin x \div 6 \sin^2 x \cos x$.

Accepted Answer

$4 \cos x \cot x$

Question 5

A city's average high temperature is modeled by $T(t) = 81.6 + 5.4 \sin(1.47t - 9.60)$, where $t$ is the month ($t=1$ is January). What is the maximum average high temperature, and in which month does it occur?

Accepted Answer

$87.0^\circ\mathrm{F}$ in July

Question 6

Which of the following is the correct double-angle identity for $	an(2x)$?

Accepted Answer

$	an(2x) = \frac{2 	an x}{1 - 	an^2 x}$

Question 7

If $\sin 	heta = -\frac{4}{5}$ and $	heta$ is in quadrant IV, what is $\sin 2	heta$?

Accepted Answer

$-\frac{24}{25}$

Question 8

Which of the following best describes the graph of $y = \csc(x)$?

Accepted Answer

Asymptotes at $x = n\pi$, $n \in \mathbb{Z}$; no zeros

Question 9

Simplify $(1 - \cos x)(1 + \cos x)$.

Accepted Answer

$\sin^2 x$

Question 10

Evaluate $\sin 62^\circ \cos 27^\circ - \cos 62^\circ \sin 27^\circ$ to four decimal places.

Accepted Answer

$\sin 35^\circ \approx 0.5736$

Question 11

Which of the following is the preferred strategy for proving a trigonometric identity?

Accepted Answer

Work on one side only until it matches the other

Question 12

Given $\sin 	heta = \frac{3}{5}$ and $	heta$ is in quadrant II, find $\sec\left(\frac{\pi}{2} - 	hetaight)$.

Accepted Answer

$-\frac{5}{3}$

Question 13

Which of the following is NOT true about the properties of trigonometric functions?

Accepted Answer

The range of tangent is $[-10, 10]$

Question 14

Simplify $\left(\cos x \div \sin x\right)^2 - \frac{1}{\sin^2 x}$.

Accepted Answer

$\cot^2 x - \csc^2 x = -1$

Question 15

Which of the following is the correct half-angle identity for $\sin\left(\frac{	heta}{2}ight)$?

Accepted Answer

$\sin\left(\frac{	heta}{2}ight) = \pm \sqrt{\frac{1 - \cos 	heta}{2}}$

Question 16

If $\sin \alpha = \frac{3}{5}$ and $\cos \beta = -\frac{5}{13}$, with $0 \leq \alpha \leq \frac{\pi}{2}$ and $\frac{\pi}{2} \leq \beta \leq \pi$, what is $\sin(\alpha + \beta)$?

Accepted Answer

$-\frac{16}{65}$

Question 17

Which of the following is the correct cofunction identity for $\sin\left(\frac{\pi}{2} - x\right)$?

Accepted Answer

$\cos x$

Question 18

Simplify $1 - 2\sin^2 x + \sin^4 x$.

Accepted Answer

$(1 - \sin^2 x)^2 = \cos^4 x$

Question 19

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

Accepted Answer

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

Precalculus Trigonometry Review – Step-by-Step Study Guidance

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