Q: Solve for $$\theta in $$the interval $$0 \leq \theta \u003c 2\pi if$$ $$\tan \theta + \sqrt{3} = 0.$$

$$\theta = \dfrac{2\pi}{3}, \dfrac{5\pi}{3}$$

Q: Verify the identity: $$\tan \theta \cos \theta = \sin \theta. $$Which of the following justifies this identity?

$$\tan \theta = \dfrac{\sin \theta}{\cos \theta}$$, so $$\tan \theta \cos \theta = \sin \theta$$

Q: Solve for $$\theta in$$ $$0 \leq \theta \u003c 2\pi$$: $$\sin(2\theta) + 1 = 0.$$

$$\theta = \dfrac{3\pi}{4}, \dfrac{7\pi}{4}$$

Q: Which of the following identities is correct?

$$(1 - \cos^2$ \theta)(1 + \cot^2$ \theta) = 1$$

Question 1

Find the exact value of $$\sin(105^\circ)$$ using the sum formula and the fact that $$105^\circ = 60^\circ + 45^\circ.$$

Accepted Answer

$$\dfrac{\sqrt{6} + \sqrt{2}}{4}$$

Question 2

Evaluate $$\cos\left(\dfrac{11\pi}{12}\right)$$ using the sum formula and the fact that $$\dfrac{11\pi}{12} = \dfrac{2\pi}{3} + \dfrac{\pi}{4}.$$

Accepted Answer

$$-\dfrac{\sqrt{6} + \sqrt{2}}{4}$$

Question 3

Solve for $$	heta in $$the interval $$0 \leq 	heta \u003c 2\pi if$$ $$	an 	heta + \sqrt{3} = 0.$$

Accepted Answer

$$	heta = \dfrac{2\pi}{3}, \dfrac{5\pi}{3}$$

Question 4

Given $$\sin x = \dfrac{1}{4}$$ and $$x is in $$quadrant I, find $$\cos(2x).$$

Accepted Answer

$$\dfrac{7}{8}$$

Question 5

If $$\cos \alpha = -\dfrac{1}{3}$$ with $$\dfrac{\pi}{2} \u003c \alpha \u003c \pi$$, and $$	an \beta = \dfrac{2}{3}$$ with $$\pi \u003c \beta \u003c \dfrac{3\pi}{2}$$, find $$\cos(\alpha - \beta). $$

Accepted Answer

$$-\dfrac{1}{3}$$

Question 6

Solve the equation $$2\sin^2$ x - 3\sin x + 1 = 0$$ for $$0 \leq x \u003c 2\pi.$$

Accepted Answer

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

Question 7

Which of the following is the general solution to $$\sin x \cos x + \sin x = 0$$?

Accepted Answer

$$x = n\pi or$$ $$x = \dfrac{\pi}{2} + 2n\pi$$, $$n \in \mathbb{Z}$$

Question 8

Verify the identity: $$	an 	heta \cos 	heta = \sin 	heta. $$Which of the following justifies this identity?

Accepted Answer

$$	an 	heta = \dfrac{\sin 	heta}{\cos 	heta}$$, so $$	an 	heta \cos 	heta = \sin 	heta$$

Question 9

Find the exact value of $$\sin$$^{-1}$$(1).$$

Accepted Answer

$$\dfrac{\pi}{2}$$

Question 10

Find the exact value of $$	an$$^{-1}$$(-1).$$

Accepted Answer

$$-\dfrac{\pi}{4}$$

Question 11

A tide in a small beach town oscillates between 7 ft (low) and 15 ft (high), with low tide at 6 AM and high tide at noon. The cycle repeats every 12 hours. Which equation models the water level $$h(t) in $$feet, where $$t is $$hours after midnight?

Accepted Answer

$$h(t) = 11 + 4\sin\left(\dfrac{\pi}{6}(t - 9)\right)$$

Question 12

Evaluate $$\sin\left(\sin$$^{-1}$$\left(\dfrac{2}{5}\right) + \cos$$^{-1}$$\left(\dfrac{2}{3}\right)\right).$$

Accepted Answer

$$\dfrac{2}{3}$$

Question 13

Solve for $$	heta in$$ $$0 \leq 	heta \u003c 2\pi$$: $$\sin(2	heta) + 1 = 0.$$

Accepted Answer

$$	heta = \dfrac{3\pi}{4}, \dfrac{7\pi}{4}$$

Question 14

Which of the following is equivalent to $$\dfrac{\sec^2$ 	heta - 1}{\sec^2$ 	heta}$$?

Accepted Answer

$$\sin^2$ 	heta$$

Question 15

Find the exact value of $$\cos$$^{-1}$$\left(\dfrac{\sqrt{2}}{2}\right).$$

Accepted Answer

$$\dfrac{\pi}{4}$$

Question 16

Which of the following identities is correct?

Accepted Answer

$$(1 - \cos^2$ 	heta)(1 + \cot^2$ 	heta) = 1$$

Question 17

Find the exact value of $$\sin$$^{-1}$$\left(\cos\left(\dfrac{\pi}{6}\right)\right).$$

Accepted Answer

$$\dfrac{\pi}{3}$$

Question 18

Solve for $$x in$$ $$0 \leq x \u003c 2\pi$$: $$2\cos^2$(3x) - 1 = 0.$$

Accepted Answer

$$x = \dfrac{\pi}{18}, \dfrac{5\pi}{18}, \dfrac{9\pi}{18}, \dfrac{13\pi}{18}, \dfrac{17\pi}{18}, \dfrac{21\pi}{18}$$

Question 19

Find the exact value of $$	an\left(\cos$$^{-1}$$\left(\dfrac{5}{13}ight)ight).$$

Accepted Answer

$$\dfrac{12}{5}$$

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