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 justifies the identity?

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

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

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

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

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

Accepted Answer

$$\dfrac{8}{15}$$

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

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

$$-\dfrac{7}{15}$$

Question 6

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

Accepted Answer

$$x = n\pi,\ x = \dfrac{\pi}{2} + n\pi$$ for integer $$n$$

Question 7

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 8

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

Accepted Answer

$$\dfrac{\pi}{6}$$

Question 9

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 10

Find the exact value: $$	an(\cos$$^{-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

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

Question 12

Find the exact value: $$\sin(2x) if$$ $$\sin x = \dfrac{1}{4}$$ and $$x is in $$quadrant I.

Accepted Answer

$$\dfrac{7}{16}$$

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: $$\sin$$^{-1}$$(1).$$

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

Which of the following identities is correct?

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

Find the exact value: $$\sin\left(\sin$$^{-1}$$\left(\dfrac{2}{5}\right) + \cos$$^{-1}$$\left(\dfrac{2}{3}\right)\right).$$

Accepted Answer

$$\dfrac{11}{15}$$

Precalculus Trigonometry Review: Step-by-Step Guidance

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