Q: Solve for all solutions to the equation $$2 - \sqrt{3} \csc(\theta) = 0. $$Which of the following is the general solution?

$$\theta = \dfrac{\pi}{3} + 2k\pi,\ \theta = \dfrac{2\pi}{3} + 2k\pi$$

Q: Which identity is established by the equation $$\csc(\theta) \cdot \cos(\theta) = \cot(\theta)$$?

True for all $$\theta$$ where defined, since $$\csc(\theta) = \dfrac{1}{\sin(\theta)}$$ and $$\cot(\theta) = \dfrac{\cos(\theta)}{\sin(\theta)}$$

Q: Which of the following identities is correct for all $$\theta$$ where defined?

$$(\sec(\theta) - 1)(\sec(\theta) + 1) = \tan^2$(\theta)$$

Q: Which of the following identities is correct?

$$\sin\left(\dfrac{\pi}{2} + \theta\right) = \cos(\theta)$$

Q: Which of the following is the correct expression for $$\tan(3\theta) in $$terms of $$\tan(\theta)$$?

$$\tan(3\theta) = \dfrac{3 \tan(\theta) - \tan^3$(\theta)}{1 - 3 \tan^2$(\theta)}$$

Q: Which of the following is the correct value for $$\sec^2$\left(\dfrac{\theta}{2}\right) in $$terms of $$\cos(\theta)$$?

$$\sec^2$\left(\dfrac{\theta}{2}\right) = \dfrac{2}{1 + \cos(\theta)}$$

Q: Which of the following identities is correct?

$$\dfrac{\sin(\theta) + \sin(3\theta)}{2 \sin(2\theta)} = \cos(\theta)$$

Question 1

Solve for all solutions to the equation $$2 - \sqrt{3} \csc(	heta) = 0. $$Which of the following is the general solution?

Accepted Answer

$$	heta = \dfrac{\pi}{3} + 2k\pi,\ 	heta = \dfrac{2\pi}{3} + 2k\pi$$

Question 2

Which of the following is the exact value of $$\cos(165^\circ)$$?

Accepted Answer

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

Question 3

Given $$\sin(	heta) = \dfrac{3}{5}$$ and $$0 \u003c 	heta \u003c \dfrac{\pi}{2}$$, what is $$\sin(2	heta)$$?

Accepted Answer

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

Question 4

Which identity is established by the equation $$\csc(	heta) \cdot \cos(	heta) = \cot(	heta)$$?

Accepted Answer

True for all $$	heta$$ where defined, since $$\csc(	heta) = \dfrac{1}{\sin(	heta)}$$ and $$\cot(	heta) = \dfrac{\cos(	heta)}{\sin(	heta)}$$

Question 5

Find the exact value of $$\sin\left(\dfrac{\pi}{12}\right) \cos\left(\dfrac{7\pi}{12}\right) - \cos\left(\dfrac{\pi}{12}\right) \sin\left(\dfrac{7\pi}{12}\right).$$

Accepted Answer

$$-\dfrac{1}{2}$$

Question 6

Which of the following is the correct half-angle formula for $$\sin(22.5^\circ)$$ and its exact value?

Accepted Answer

$$\sin(22.5^\circ) = \sqrt{\dfrac{1 - \cos(45^\circ)}{2}} = \dfrac{\sqrt{2 - \sqrt{2}}}{2}$$

Question 7

Which of the following identities is correct for all $$	heta$$ where defined?

Accepted Answer

$$(\sec(	heta) - 1)(\sec(	heta) + 1) = 	an^2$(	heta)$$

Question 8

Given $$	an(	heta) = \dfrac{4}{3}$$ and $$\pi \u003c 	heta \u003c \dfrac{3\pi}{2}$$, what is $$\cos(2	heta)$$?

Accepted Answer

$$-\dfrac{7}{25}$$

Question 9

Which of the following is the correct value for $$\cos(70^\circ) \cos(20^\circ) - \sin(70^\circ) \sin(20^\circ)$$?

Accepted Answer

$$\cos(90^\circ) = 0$$

Question 10

Which of the following identities is correct?

Accepted Answer

$$\sin\left(\dfrac{\pi}{2} + 	hetaight) = \cos(	heta)$$

Question 11

Which of the following is the correct expression for $$	an(3	heta) in $$terms of $$	an(	heta)$$?

Accepted Answer

$$	an(3	heta) = \dfrac{3 	an(	heta) - 	an^3$(	heta)}{1 - 3 	an^2$(	heta)}$$

Question 12

Which of the following is the correct value for $$\sec^2$\left(\dfrac{	heta}{2}ight) in $$terms of $$\cos(	heta)$$?

Accepted Answer

$$\sec^2$\left(\dfrac{	heta}{2}ight) = \dfrac{2}{1 + \cos(	heta)}$$

Question 13

Which of the following identities is correct?

Accepted Answer

$$\dfrac{\sin(	heta) + \sin(3	heta)}{2 \sin(2	heta)} = \cos(	heta)$$

Question 14

Solve the equation $$2 \sin^2$(	heta) - \sin(	heta) - 1 = 0$$ for $$0 \leq 	heta \u003c 2\pi. $$Which of the following is the correct set of solutions?

Accepted Answer

Question 15

Which of the following is the correct value for $$\cos\left(\dfrac{7\pi}{12}\right)$$?

Accepted Answer

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

Question 16

Which of the following is the correct value for $$\sin\left(\dfrac{\pi}{2} + 	hetaight)$$?

Accepted Answer

$$\cos(	heta)$$

Question 17

Which of the following is the correct value for $$\cos(	heta)[	an(	heta) + \cot(	heta)]$$?

Accepted Answer

$$\csc(	heta)$$

Question 18

Which of the following is the correct value for $$\sin(\alpha + \beta) / [\sin(\alpha) \cos(\beta)]$$?

Accepted Answer

$$1 + \cot(\alpha) 	an(\beta)$$

Precalculus Trigonometric Equations and Identities Study Guide

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