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

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

Question 1

Which of the following is equivalent to $$\sin 	heta \cos 	heta + \cos 	heta \sin 	heta$$?

Accepted Answer

$$\sin(2	heta)$$

Question 2

Given $$\cos 	heta = -\frac{3}{5}$$ and $$	heta is in $$Quadrant III, what is $$\sin(2	heta)$$?

Accepted Answer

$$-\frac{24}{25}$$

Question 3

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

Accepted Answer

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

Question 4

If $$	an 	heta = \frac{3}{4}$$ and $$	heta is in $$Quadrant III, what is $$\cos 	heta$$?

Accepted Answer

$$-\frac{4}{5}$$

Question 5

Which of the following is a Pythagorean identity?

Accepted Answer

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

Question 6

A Ferris wheel with a radius of 20 meters completes one revolution every 2 minutes. The bottom of the wheel is 2 meters above the ground. Which function models the height $$h(t) (in $$meters) of a passenger above the ground $$t$$ minutes after starting at the lowest point?

Accepted Answer

$$h(t) = 20 \cos(\pi t) + 22$$

Question 7

Simplify $$\frac{	an x + 	an y}{1 - 	an x 	an y}.$$

Accepted Answer

$$	an(x + y)$$

Question 8

The voltage $$V in an $$electrical circuit is $$V = 120 \sin(120\pi t). At $$what smallest positive time $$t (in $$seconds) is the voltage $$60$$?

Accepted Answer

$$\frac{1}{720}$$

Question 9

Which of the following is equivalent to $$\cos(3x)$$?

Accepted Answer

$$4 \cos^3$ x - 3 \cos x$$

Question 10

Find the exact value of $$	an\left[\arccos\left(-\frac{1}{2}ight)ight].$$

Accepted Answer

$$-\sqrt{3}$$

Question 11

A rain gutter is constructed from a sheet of aluminum 12 inches wide by bending up one-third of the sheet on each side through an angle $$	heta. $$The area $$A of $$the opening is given by $$A = 16 \sin 	heta (1 + \cos 	heta). $$For what value of $$	heta \in (0, \frac{\pi}{2}) is $$the area $$A = 12\sqrt{3}$$?

Accepted Answer

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

Question 12

Which of the following is the range of $$f(x) = \arcsin x$$?

Accepted Answer

$$[-\frac{\pi}{2}, \frac{\pi}{2}]$$

Question 13

Express $$\sin(x - \frac{\pi}{2}) as a $$cofunction.

Accepted Answer

$$-\cos x$$

Question 14

If $$\sin \alpha = \frac{4}{5}$$ with $$\alpha in $$Quadrant II and $$\cos \beta = -\frac{5}{13}$$ with $$\beta in $$Quadrant III, what is $$\sin(\alpha - \beta)$$?

Accepted Answer

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

Question 15

Which of the following is equivalent to $$\frac{1 - \cos(2	heta)}{2}$$?

Accepted Answer

$$\sin^2$ 	heta$$

Question 16

Given $$\cos 	heta = 0.3420$$ and $$	heta is in $$Quadrant I, find $$\sin\left(\frac{	heta}{2}ight).$$

Accepted Answer

$$0.8192$$

Question 17

Which of the following is the correct sum-to-product formula for $$\sin A + \sin B$$?

Accepted Answer

$$2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)$$

Question 18

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

Accepted Answer

$$\frac{\pi}{2}, \frac{7\pi}{6}, \frac{11\pi}{6}$$

Question 19

The horizontal distance $$d a $$soccer ball travels is $$d = 20 \sin 	heta \cos 	heta. $$Using a double-angle identity, rewrite this model.

Accepted Answer

$$d = 10 \sin(2	heta)$$

Question 20

Application: In the study of projectiles, the range $$R is $$given by $$R = \frac{v_0$^2 \sin(2	heta)}{g}. $$For what angle $$	heta is $$the range maximized?

Accepted Answer

$$45^\circ$$

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