Q: Given $$\cos(67.5^\circ)$$ and $$67.5^\circ = \dfrac{135^\circ}{2}$$, which identity should you use to find its exact value, and what is the result?

Use the half-angle identity: $$\cos(\theta/2) = \pm\sqrt{\dfrac{1 + \cos\theta}{2}}$$; $$\cos(67.5^\circ) = \dfrac{\sqrt{2 + \sqrt{2}}}{2}$$

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

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

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

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

Q: Which of the following identities is correct?

$$\dfrac{\sec^2$ \theta - 1}{\sec^2$ \theta} = \sin^2$ \theta$$

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

$$\theta = \dfrac{3\pi}{4}, \dfrac{5\pi}{4}$$

Question 1

Which of the following is the correct exact value for $$\sin(105^\circ)$$ 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 $$67.5^\circ = \dfrac{135^\circ}{2}$$, which identity should you use to find its exact value, and what is the result?

Accepted Answer

Use the half-angle identity: $$\cos(	heta/2) = \pm\sqrt{\dfrac{1 + \cos	heta}{2}}$$; $$\cos(67.5^\circ) = \dfrac{\sqrt{2 + \sqrt{2}}}{2}$$

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

Given $$\cos \alpha = -\dfrac{1}{3}$$ with $$\dfrac{\pi}{2} \u003c \alpha \u003c \pi$$, and $$	an \beta = \dfrac{2}{3}$$ with $$\pi \u003c \beta \u003c \dfrac{3\pi}{2}$$, what is $$\cos(\alpha - \beta)$$?

Accepted Answer

$$-\dfrac{5}{9}$$

Question 8

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

Accepted Answer

$$x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}$$

Question 9

Which of the following is the correct value for $$\sin$$^{-1}$$(1)$$?

Accepted Answer

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

Question 10

A beach town's tide oscillates between 7 ft (low) and 15 ft (high), with low tide at 6 AM and high tide at noon, repeating every 12 hours. Which equation models the water level $$h(t) in $$feet, where $$t is $$hours after midnight?

Accepted Answer

$$h(t) = 4\sin\left(\dfrac{\pi}{12}(t - 9)\right) + 11$$

Question 11

Which of the following is the correct value for $$\cos$$^{-1}$$\left(\dfrac{\sqrt{2}}{2}\right)$$?

Accepted Answer

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

Question 12

Which of the following identities is correct?

Accepted Answer

$$\dfrac{\sec^2$ 	heta - 1}{\sec^2$ 	heta} = \sin^2$ 	heta$$

Question 13

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

Accepted Answer

$$	heta = \dfrac{3\pi}{4}, \dfrac{5\pi}{4}$$

Question 14

Which of the following is the correct value for $$	an$$^{-1}$$(-1)$$?

Accepted Answer

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

Question 15

If $$\sin x = \dfrac{x}{x+1}$$, what is $$\cos(\sin$$^{-1}$$(\dfrac{x}{x+1}))$$?

Accepted Answer

$$\dfrac{1}{x+1}$$

Question 16

Which of the following is the correct value for $$\sin$$^{-1}$$(\cos(\dfrac{\pi}{6}))$$?

Accepted Answer

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

Question 17

Which of the following is the correct value for $$	an(\cos$$^{-1}$$(\dfrac{5}{13}))$$?

Accepted Answer

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

Question 18

Which of the following is the correct value for $$2\sin\left(\dfrac{\pi}{12}\right)\cos\left(\dfrac{\pi}{12}\right)$$?

Accepted Answer

$$\sin\left(\dfrac{\pi}{6}\right)$$

Question 19

Which of the following is the correct value for $$\cos\left(	an$$^{-1}$$\left(\dfrac{1}{2}ight) + \sin$$^{-1}$$(-1)ight)$$?

Accepted Answer

$$0$$

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