Question 1

Simplify the expression $$\displaystyle \frac{1 + \cos(2\omega)}{\cos \omega}.$$

Accepted Answer

$$2 \cos \omega$$

Question 2

Which of the following is equivalent to $$\csc^2$ x - 1$$?

Accepted Answer

$$\cot^2$ x$$

Question 3

Simplify $$\displaystyle \frac{\sin^2$ x + \cos^2$ x}{1 + \cot^2$ x}.$$

Accepted Answer

$$\sin^2$ x$$

Question 4

Evaluate $$\sin(45^\circ + 30^\circ).$$

Accepted Answer

$$\displaystyle \frac{\sqrt{6} + \sqrt{2}}{4}$$

Question 5

Evaluate $$	an(60^\circ - 45^\circ).$$

Accepted Answer

$$\displaystyle \frac{\sqrt{3} - 1}{1 + \sqrt{3}}$$

Question 6

Simplify $$\cos(x + 180^\circ) + \cos x.$$

Accepted Answer

$$0$$

Question 7

Rewrite $$\sin x \cos 45^\circ + \cos x \sin 45^\circ as a $$single trigonometric function.

Accepted Answer

$$\sin(x + 45^\circ)$$

Question 8

Evaluate $$\sin(2 	imes 75^\circ).$$

Accepted Answer

$$\sin(150^\circ) = \displaystyle \frac{1}{2}$$

Question 9

Evaluate $$\cos(15^\circ)$$ using a sum or difference identity.

Accepted Answer

$$\displaystyle \frac{\sqrt{6} + \sqrt{2}}{4}$$

Question 10

Solve for $$x in$$ $$[0, 2\pi)$$: $$2 \sin x = \sqrt{3}.$$

Accepted Answer

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

Question 11

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

Accepted Answer

$$x = \displaystyle \frac{\pi}{2}, \frac{7\pi}{6}, \frac{11\pi}{6}$$

Question 12

Solve for $$x in$$ $$[0, 2\pi)$$: $$	an(3x) = 1.$$

Accepted Answer

$$x = \displaystyle \frac{\pi}{12}, \frac{5\pi}{12}, \frac{9\pi}{12}$$

Question 13

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

Accepted Answer

$$x = \displaystyle \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$$

Question 14

A Ferris wheel makes one complete revolution every 2 minutes. If the height $$h (in $$meters) of a seat above the ground at time $$t (in $$minutes) is modeled by $$h(t) = 10 + 8 \sin(\pi t)$$, what is the maximum height reached by the seat?

Accepted Answer

$$18$$ meters

Question 15

If $$	an(2	heta) = \sqrt{3}$$ and $$0 \u003c 	heta \u003c \frac{\pi}{2}$$, what is $$	heta$$?

Accepted Answer

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

Question 16

Which of the following is the correct value for $$\sin(135^\circ)$$?

Accepted Answer

$$\displaystyle \frac{\sqrt{2}}{2}$$

Question 17

A sound wave can be modeled by $$y = 3 \sin(2\pi t)$$, where $$t is in $$seconds. What is the amplitude and period of the wave?

Accepted Answer

Amplitude $$= 3$$, Period $$= 1$$ second

Question 18

If $$\sin 	heta = \displaystyle \frac{1}{2}$$ and $$0 \u003c 	heta \u003c \pi$$, what are the possible values of $$	heta$$?

Accepted Answer

$$\displaystyle \frac{\pi}{6}, \frac{5\pi}{6}$$

Question 19

A ladder leans against a wall making an angle of $$60^\circ$$ with the ground. If the ladder is $$10$$ meters long, how high up the wall does it reach?

Accepted Answer

$$10 \sin 60^\circ = 5\sqrt{3}$$ meters

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