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

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

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