Q: Which of the following is a correct statement of the first Pythagorean trigonometric identity?

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

Q: Which of the following is NOT an identity for all $$\theta$$?

$$\sin \theta = \sqrt{1 - \cos^2$ \theta}$$ \,\text{for}\, \theta = \frac{3\pi}{2}$

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

$$\sin 2\theta = 2 \sin \theta \cos \theta$$

Q: Which of the following is the correct half-angle formula for $$\sin^2$ \theta$$?

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

Q: Which of the following is the correct power reduction formula for $$\cos^2$ \theta$$?

$$\cos^2$ \theta = \frac{1 + \cos 2\theta}{2}$$

Q: Which of the following is the correct identity for $$\tan^2$ \theta + 1$$?

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

Q: Which of the following is the correct expression for $$\cos(2\theta) in $$terms of $$\sin^2$ \theta$$?

$$\cos(2\theta) = 1 - 2\sin^2$ \theta$$

Question 1

Which of the following is a correct statement of the first Pythagorean trigonometric identity?

Accepted Answer

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

Question 2

If $$\cos 	heta = -\frac{12}{13}$$ and $$	an 	heta \u003c 0$$, what is $$\sin 2	heta$$?

Accepted Answer

$$\frac{120}{169}$$

Question 3

Which of the following is NOT an identity for all $$	heta$$?

Accepted Answer

$$\sin 	heta = \sqrt{1 - \cos^2$ 	heta}$$ \,	ext{for}\, 	heta = \frac{3\pi}{2}$

Question 4

Given $$	an u = \frac{3}{4}$$ with $$u in $$quadrant I and $$\sin v = \frac{5}{13}$$ with $$v in $$quadrant II, what is $$\sin(u+v)$$?

Accepted Answer

$$\frac{1}{13}$$

Question 5

Which formula correctly expresses $$\cos(\alpha + \beta) in $$terms of $$\cos \alpha$$, $$\cos \beta$$, $$\sin \alpha$$, and $$\sin \beta$$?

Accepted Answer

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

Question 6

If $$\sec 	heta = -\sqrt{3}$$ and $$\sin 	heta \u003e 0$$, what is $$	an 2	heta$$?

Accepted Answer

$$-2\sqrt{3}$$

Question 7

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

Accepted Answer

$$\sin 2	heta = 2 \sin 	heta \cos 	heta$$

Question 8

Which of the following is a valid trigonometric substitution for $$y = \sqrt{1 - x^2$}$$?

Accepted Answer

$$x = \sin 	heta$$

Question 9

Find the exact value of $$\sin(15^\circ)$$ using the sum and difference formulas.

Accepted Answer

$$\frac{\sqrt{6} - \sqrt{2}}{4}$$

Question 10

Which of the following is the correct half-angle formula for $$\sin^2$ 	heta$$?

Accepted Answer

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

Question 11

If $$\cos 	heta = -\frac{12}{13}$$ and $$	an 	heta \u003c 0$$, what is $$	an 2	heta$$?

Accepted Answer

$$\frac{5}{12}$$

Question 12

Which of the following is the correct power reduction formula for $$\cos^2$ 	heta$$?

Accepted Answer

$$\cos^2$ 	heta = \frac{1 + \cos 2	heta}{2}$$

Question 13

Which of the following is the correct identity for $$	an^2$ 	heta + 1$$?

Accepted Answer

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

Question 14

If $$y = \sqrt{x^2$ - 4}$$, which substitution can be used to rewrite $$y in $$terms of $$	heta$$?

Accepted Answer

$$x = 2 \sec 	heta$$

Question 15

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

Accepted Answer

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

Question 16

Which of the following is the correct expression for $$\cos(2	heta) in $$terms of $$\sin^2$ 	heta$$?

Accepted Answer

$$\cos(2	heta) = 1 - 2\sin^2$ 	heta$$

Question 17

Which of the following is the correct expression for $$\cos(2	heta) in $$terms of $$\cos^2$ 	heta$$?

Accepted Answer

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

Question 18

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

Accepted Answer

$$	an(\alpha + \beta) = \frac{	an \alpha + 	an \beta}{1 - 	an \alpha 	an \beta}$$

Trigonometric Identities and Formulas: A Precalculus Study Guide

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