Q: A medical imaging device uses the double-angle identity for sine. If $$\sin(\theta) = 0.6$$ and $$\cos(\theta) = 0.8$$, what is $$\sin(2\theta)$$?

$$2 \times 0.6 \times 0.8 = 0.96$$

Q: A chemist is analyzing a reaction that oscillates with a period related to $$\tan(2\theta). If$$ $$\tan \theta = 3$$, what is $$\tan(2\theta)$$?

$$\dfrac{2 \times 3}{1 - 3^2$} = \dfrac{6}{-8} = -\dfrac{3}{4}$$

Q: Which of the following is the correct sum identity for $$\tan(\alpha + \beta)$$?

$$\dfrac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$$

Question 1

Which of the following is a correct form of the Pythagorean Identity for trigonometric functions?

Accepted Answer

$$\sin^2$ \alpha + \cos^2$ \alpha = 1$$

Question 2

Which trigonometric function is classified as an even function?

Accepted Answer

$$\cos(x)$$

Question 3

Given the identity $$	an(\alpha) = \dfrac{\sin(\alpha)}{\cos(\alpha)}$$, which of the following is equivalent to $$\cot(\alpha)$$?

Accepted Answer

$$\dfrac{\cos(\alpha)}{\sin(\alpha)}$$

Question 4

Which of the following identities expresses $$\cos(2\alpha) in $$terms of $$\sin^2$ \alpha$$ only?

Accepted Answer

$$1 - 2\sin^2$ \alpha$$

Question 5

A sound engineer is analyzing a wave and needs to express $$\sin(75^\circ) in $$terms of $$\sin(45^\circ)$$ and $$\cos(30^\circ). $$Which identity should be used?

Accepted Answer

$$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$$

Question 6

Which of the following is the correct cofunction identity for $$\cos\left(\dfrac{\pi}{2} - \alpha\right)$$?

Accepted Answer

$$\sin(\alpha)$$

Question 7

A medical imaging device uses the double-angle identity for sine. If $$\sin(	heta) = 0.6$$ and $$\cos(	heta) = 0.8$$, what is $$\sin(2	heta)$$?

Accepted Answer

$$2 	imes 0.6 	imes 0.8 = 0.96$$

Question 8

Which of the following is NOT a valid strategy for verifying trigonometric identities?

Accepted Answer

Multiply both sides by zero to simplify

Question 9

Solve for $$x in$$ $$[0, 2\pi)$$: $$\sin x = \dfrac{1}{2}.$$

Accepted Answer

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

Question 10

Which of the following is the correct half-angle identity for $$\sin\left(\dfrac{\alpha}{2}\right)$$?

Accepted Answer

$$\pm \sqrt{\dfrac{1 - \cos \alpha}{2}}$$

Question 11

A chemist is analyzing a reaction that oscillates with a period related to $$	an(2	heta). If$$ $$	an 	heta = 3$$, what is $$	an(2	heta)$$?

Accepted Answer

$$\dfrac{2 	imes 3}{1 - 3^2$} = \dfrac{6}{-8} = -\dfrac{3}{4}$$

Question 12

Which of the following is a valid solution for $$\cos \beta = -\dfrac{4}{5} in$$ $$[0, 2\pi)$$?

Accepted Answer

$$\beta = 2.498 or$$ $$\beta = 3.785$$

Question 13

Which of the following is the correct sum identity for $$	an(\alpha + \beta)$$?

Accepted Answer

$$\dfrac{	an \alpha + 	an \beta}{1 - 	an \alpha 	an \beta}$$

Question 14

A satellite dish is designed using the identity $$\cos(2\alpha) = 2\cos^2$ \alpha - 1. If$$ $$\cos \alpha = 0.7$$, what is $$\cos(2\alpha)$$?

Accepted Answer

$$2 	imes 0.7^2$ - 1 = 2 	imes 0.49 - 1 = 0.98 - 1 = -0.02$$

Question 15

Which of the following is the correct reciprocal identity for $$\sec \alpha$$?

Accepted Answer

$$\dfrac{1}{\cos \alpha}$$

Question 16

Solve for $$	heta in$$ $$[0, 2\pi)$$: $$3	an 	heta - 2 = 10.$$

Accepted Answer

$$	heta = 1.326,\ 4.467$$

Question 17

Which of the following is the correct expression for $$	an\left(\dfrac{\alpha}{2}ight) in $$terms of $$\sin \alpha$$ and $$\cos \alpha$$?

Accepted Answer

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

Question 18

A physicist is analyzing a pendulum and needs to verify the identity $$\sin(2	heta) = 2\sin 	heta \cos 	heta. $$Which of the following is a correct application of this identity if $$\sin 	heta = 0.5$$ and $$\cos 	heta = 0.866$$?

Accepted Answer

$$2 	imes 0.5 	imes 0.866 = 0.866$$

Trigonometric Identities and Conditional Equations: A Precalculus Study Guide

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