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

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

Q: Which of the following is a correct application of the Pythagorean identity?

$$1 + \tan^2$\theta = \sec^2$\theta$$

Q: If $$\cos 2\theta = 1 - 2\sin^2$\theta$$, what is $$\sin^2$\theta in $$terms of $$\cos 2\theta$$?

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

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

$$\cos\left(\dfrac{\theta}{2}\right) = \pm \sqrt{\dfrac{1 + \cos\theta}{2}}$$

Q: A satellite dish is mounted so that its angle of elevation $$\theta$$ satisfies $$\sin 2\theta = 0.8. $$What is one possible value of $$\theta in$$ $$[0, \pi)$$?

$$\theta = \dfrac{1}{2} \arcsin(0.8)$$

Question 1

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

Accepted Answer

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

Question 2

Given $$\sin \alpha = \dfrac{3}{5}$$ with $$\alpha in $$Quadrant II, and $$\cos \beta = -\dfrac{5}{13}$$ with $$\beta in $$Quadrant III, what is $$\sin(\alpha + \beta)$$?

Accepted Answer

$$\sin(\alpha + \beta) = -\dfrac{56}{65}$$

Question 3

Which of the following is a correct application of the Pythagorean identity?

Accepted Answer

$$1 + 	an^2$	heta = \sec^2$	heta$$

Question 4

If $$\cos 2	heta = 1 - 2\sin^2$	heta$$, what is $$\sin^2$	heta in $$terms of $$\cos 2	heta$$?

Accepted Answer

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

Question 5

Solve for all $$x in$$ $$[0, 2\pi)$$: $$2\sin^2$ x + 3\sin x + 1 = 0.$$

Accepted Answer

$$x = \dfrac{\pi}{2}, \dfrac{3\pi}{2}, \dfrac{7\pi}{6}, \dfrac{11\pi}{6}$$

Question 6

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

Accepted Answer

$$\cos\left(\dfrac{	heta}{2}ight) = \pm \sqrt{\dfrac{1 + \cos	heta}{2}}$$

Question 7

A satellite dish is mounted so that its angle of elevation $$	heta$$ satisfies $$\sin 2	heta = 0.8. $$What is one possible value of $$	heta in$$ $$[0, \pi)$$?

Accepted Answer

$$	heta = \dfrac{1}{2} \arcsin(0.8)$$

Question 8

Given $$	an \alpha = -2$$ with $$\alpha in $$Quadrant IV, find $$\sin 2\alpha.$$

Accepted Answer

$$\sin 2\alpha = -\dfrac{4}{5}$$

Question 9

Which of the following is the correct general solution for $$\cos x = \dfrac{1}{2}$$?

Accepted Answer

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

Question 10

If $$\sin x = -\dfrac{1}{2}$$, what are all solutions for $$x in$$ $$[0, 2\pi)$$?

Accepted Answer

$$x = \dfrac{7\pi}{6}, \dfrac{11\pi}{6}$$

Question 11

Which of the following is the correct sum formula for $$\sin(A+B)$$?

Accepted Answer

$$\sin(A+B) = \sin A \cos B + \cos A \sin B$$

Question 12

A medical device rotates through an angle $$x$$ such that $$\cos 4x = -\dfrac{1}{2}. $$Find all solutions for $$x in$$ $$[0, 2\pi).$$

Accepted Answer

$$x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{11\pi}{6}$$

Question 13

Which of the following is the correct even/odd property for $$	an(-x)$$?

Accepted Answer

$$	an(-x) = -	an x$$

Question 14

If $$\sin^2$ x - \dfrac{1}{4} = 0$$, what are all solutions for $$x in$$ $$[0, 2\pi)$$?

Accepted Answer

$$x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{11\pi}{6}$$

Question 15

A chemical reaction's rate is modeled by $$R = \sin(	heta)\cos(	heta). $$Which identity can be used to rewrite $$R in $$terms of $$\sin 2	heta$$?

Accepted Answer

$$R = \dfrac{1}{2}\sin 2	heta$$

Question 16

Given $$\sin(	heta/2) = \dfrac{1}{\sqrt{2}}$$ and $$	heta/2 is in $$Quadrant I, what is $$	heta$$?

Accepted Answer

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

Question 17

Which of the following is the correct reciprocal identity for $$\csc 	heta$$?

Accepted Answer

$$\csc 	heta = \dfrac{1}{\sin 	heta}$$

Question 18

If $$	an \alpha = 3$$ and $$	an \beta = 2$$, what is $$	an(\alpha + \beta)$$?

Accepted Answer

$$	an(\alpha + \beta) = \dfrac{3 + 2}{1 - 3 \cdot 2} = -1$$

Trigonometric Identities and Equations: Study Guide

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