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

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

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

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

Q: Solve for $$\theta in$$ $$0 \leq \theta \u003c 2\pi$$: $$\cos \theta = \dfrac{1}{2}.$$

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

Q: Which of the following is the correct verification for the identity $$\dfrac{\sec^2$ \theta - 1}{\sec^2$ \theta} = \sin^2$ \theta$$?

Because $$\sec^2$ \theta - 1 = \tan^2$ \theta$$ and $$\sec^2$ \theta = 1 + \tan^2$ \theta$$, so $$\dfrac{\tan^2$ \theta}{1 + \tan^2$ \theta} = \sin^2$ \theta$$

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

Given $$\cos(67.5^\circ)$$ and $$67.5^\circ = \dfrac{135^\circ}{2}$$, find the exact value using the half-angle formula.

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

If $$\sin x = \dfrac{1}{4}$$ and $$x is in $$quadrant I, what is $$\cos(2x)$$?

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

Given $$\cos \alpha = -\dfrac{1}{3}$$ 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 9

Find 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 10

Evaluate $$	an(\cos$$^{-1}$$(\dfrac{5}{13})).$$

Accepted Answer

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

Question 11

Which equation models the water level $$h(t) (in $$feet) at time $$t (in $$hours after midnight) for a tide that oscillates between 7 ft and 15 ft, with low tide at 6 AM and high tide at noon, repeating every 12 hours?

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

Solve for $$	heta in$$ $$0 \leq 	heta \u003c 2\pi$$: $$\cos 	heta = \dfrac{1}{2}.$$

Accepted Answer

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

Question 14

Which of the following is the correct verification for the identity $$\dfrac{\sec^2$ 	heta - 1}{\sec^2$ 	heta} = \sin^2$ 	heta$$?

Accepted Answer

Because $$\sec^2$ 	heta - 1 = 	an^2$ 	heta$$ and $$\sec^2$ 	heta = 1 + 	an^2$ 	heta$$, so $$\dfrac{	an^2$ 	heta}{1 + 	an^2$ 	heta} = \sin^2$ 	heta$$

Question 15

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

Accepted Answer

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

Question 16

If $$\sin x = \dfrac{1}{4}$$ and $$x is in $$quadrant I, what is $$	an(2x)$$?

Accepted Answer

$$\dfrac{4}{3}$$

Question 17

Evaluate $$\sin(2	heta) + \sqrt{3}\cos 	heta = 0$$ for $$0 \leq 	heta \u003c 2\pi. $$Which is a solution?

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

$$\dfrac{1}{2}\sin(2x) = 2\sin x \cos x = \dfrac{\sqrt{15}}{8}$$

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