Q: Solve for $$\theta in$$ $$0 \leq \theta \u003c 2\pi$$: $$2\sin\theta + 3 = 2. $$What is the solution?

$$\theta = \dfrac{\pi}{6}, \dfrac{5\pi}{6}$$

Q: Solve for $$\theta in$$ $$0 \leq \theta \u003c 2\pi$$: $$4\cos^2$\theta - 3 = 0. $$What are the solutions?

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

Q: Which of the following is the general solution for $$\sin(2\theta) = -1$$?

$$2\theta = \dfrac{3\pi}{2} + 2\pi n$$

Q: Which of the following is the correct half-angle formula for $$\tan\left(\dfrac{\theta}{2}\right)$$?

$$\tan\left(\dfrac{\theta}{2}\right) = \dfrac{1 - \cos\theta}{\sin\theta}$$

Q: Which of the following is the correct double-angle formula for $$\cos 2\theta$$?

$$\cos 2\theta = 2\cos^2$\theta - 1$$

Question 1

Find the exact value of $$\sin(-15^\circ). $$Which of the following is correct?

Accepted Answer

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

Question 2

Given $$\cos \alpha = \dfrac{3}{5}$$ with $$0 \u003c \alpha \u003c \dfrac{\pi}{2}$$ and $$\sin \beta = -\dfrac{4}{5}$$ with $$-\dfrac{\pi}{2} \u003c \beta \u003c 0$$, find $$\cos(\alpha + \beta). $$Which is correct?

Accepted Answer

$$\dfrac{24}{25}$$

Question 3

Find the exact value of $$	an\left(\dfrac{7\pi}{12}ight). $$Which is correct?

Accepted Answer

$$2 - \sqrt{3}$$

Question 4

Given $$\cos 	heta = -\dfrac{3}{5}$$ with $$\dfrac{\pi}{2} \u003c 	heta \u003c \pi$$, find $$\sin\left(\dfrac{	heta}{2}ight). $$Which is correct?

Accepted Answer

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

Question 5

Which of the following is a correct identity?

Accepted Answer

$$\dfrac{1 + \sin \beta}{1 - \sin \beta} = \dfrac{\csc \beta + 1}{\csc \beta - 1}$$

Question 6

Solve for $$	heta in$$ $$0 \leq 	heta \u003c 2\pi$$: $$2\sin	heta + 3 = 2. $$What is the solution?

Accepted Answer

$$	heta = \dfrac{\pi}{6}, \dfrac{5\pi}{6}$$

Question 7

Establish the identity: $$\cos 	heta \left(1 + \sin 	hetaight) + \dfrac{1 + \sin 	heta}{\cos 	heta} = 2\sec 	heta. $$Which of the following steps is correct?

Accepted Answer

Combine terms over a common denominator and simplify to $$2\sec 	heta$$

Question 8

Solve for $$	heta in$$ $$0 \leq 	heta \u003c 2\pi$$: $$4\cos^2$	heta - 3 = 0. $$What are the solutions?

Accepted Answer

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

Question 9

Which of the following is the general solution for $$\sin(2	heta) = -1$$?

Accepted Answer

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

Question 10

Given $$	an A + 	an B = \cot A + \cot B$$, which of the following is equivalent to $$	an A 	an B$$?

Accepted Answer

$$	an A 	an B$$

Question 11

Solve for $$	heta in$$ $$0 \leq 	heta \u003c 2\pi$$: $$\cos^2$	heta - \sin^2$	heta + \sin	heta = 0. $$Which is a solution?

Accepted Answer

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

Question 12

Which of the following is the correct half-angle formula for $$	an\left(\dfrac{	heta}{2}ight)$$?

Accepted Answer

$$	an\left(\dfrac{	heta}{2}ight) = \dfrac{1 - \cos	heta}{\sin	heta}$$

Question 13

If $$	an\left(\dfrac{	heta}{2} + \dfrac{\pi}{3}ight) = 1$$, what is one possible value for $$	heta in$$ $$0 \leq 	heta \u003c 2\pi$$?

Accepted Answer

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

Question 14

Which of the following is the correct double-angle formula for $$\cos 2	heta$$?

Accepted Answer

$$\cos 2	heta = 2\cos^2$	heta - 1$$

Question 15

Solve for $$	heta in$$ $$0 \leq 	heta \u003c 2\pi$$: $$\cos	heta + \sin	heta = 0. $$Which is a solution?

Accepted Answer

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

Question 16

Which of the following is the correct expression for $$\sec \mu - 	an \mu$$?

Accepted Answer

$$\dfrac{\cos \mu}{1 + \sin \mu}$$

Question 17

Find the exact value of $$	an\left(\dfrac{5\pi}{8}ight). $$Which is correct?

Accepted Answer

$$2 + \sqrt{3}$$

Question 18

Which of the following is the correct identity for $$1 + 	an 	heta \cot 	heta$$?

Accepted Answer

$$1$$

Question 19

Solve for $$	heta in$$ $$0 \leq 	heta \u003c 2\pi$$: $$\sin(2	heta)\sin	heta = \cos	heta. $$Which is a solution?

Accepted Answer

$$	heta = 0$$

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