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

$$\theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}$$

$$\cos \theta + \frac{3\pi}{2} = \sin \theta$$

Question 1

Which of the following is the correct application of the reciprocal identity to rewrite $$\sec(-x) \sin x in $$terms of sine and cosine?

Accepted Answer

$$\sin x \cdot \frac{1}{\cos(-x)}$$

Question 2

Given the identity $$\sec(-x) \sin x = 	an x$$, which step correctly applies the even-odd identity?

Accepted Answer

$$\cos(-x) = \cos x$$

Question 3

Which of the following expresses $$\sin 80^\circ \cos 10^\circ + \cos 80^\circ \sin 10^\circ as a $$single trigonometric function?

Accepted Answer

$$\sin(80^\circ + 10^\circ)$$

Question 4

Find the exact value of $$\sin 80^\circ \cos 10^\circ + \cos 80^\circ \sin 10^\circ.$$

Accepted Answer

$$1$$

Question 5

Which of the following is the correct double-angle identity for $$2 \sin 75^\circ \cos 75^\circ$$?

Accepted Answer

$$\sin 150^\circ$$

Question 6

What is the exact value of $$2 \sin 75^\circ \cos 75^\circ$$?

Accepted Answer

$$\frac{1}{2}$$

Question 7

Use a half-angle formula to find the exact value of $$\sin 112.5^\circ. $$Which formula and angle should be used?

Accepted Answer

$$\sin 112.5^\circ = \sqrt{\frac{1 - \cos 225^\circ}{2}}$$

Question 8

Express $$\sin 345^\circ + \sin 255^\circ as a $$product using a sum-to-product identity.

Accepted Answer

$$2 \sin 300^\circ \cos 45^\circ$$

Question 9

Verify the identity: $$\frac{\sin(3x) - \sin(5x)}{\cos(3x) - \cos(5x)} = -\cot(4x). $$Which identity is used in the final simplification?

Accepted Answer

Quotient Identity

Question 10

Solve for $$	heta in$$ $$5 \sin 	heta - 1 = 3 \sin 	heta$$ for $$0 \leq 	heta \u003c 2\pi.$$

Accepted Answer

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

Question 11

Solve $$\sin(2	heta) = \frac{\sqrt{3}}{2}$$ for $$	heta in$$ $$[0, 2\pi).$$

Accepted Answer

$$	heta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}$$

Question 12

Use an identity to solve $$2 \sin x - \cos x - 1 = 0$$ for $$x in$$ $$[0, 2\pi).$$

Accepted Answer

$$x = \frac{\pi}{3}, \frac{5\pi}{3}$$

Question 13

Solve $$\cos 2x = -\cos x$$ for $$x in$$ $$[0, 2\pi).$$

Accepted Answer

$$x = \frac{\pi}{3}, \frac{5\pi}{3}$$

Question 14

Which of the following is the correct half-angle formula for $$\cos 67.5^\circ$$?

Accepted Answer

$$\cos 67.5^\circ = \sqrt{\frac{1 + \cos 135^\circ}{2}}$$

Question 15

Which of the following identities expresses $$\cos 	heta + \frac{3\pi}{2} in $$terms of $$\sin 	heta$$?

Accepted Answer

$$\cos 	heta + \frac{3\pi}{2} = \sin 	heta$$

Question 16

Write $$\cos 180^\circ - \sin 180^\circ as a $$double-angle function and find its exact value.

Accepted Answer

$$\cos 360^\circ = 1$$

Question 17

Which of the following is the correct sum-to-product formula for $$\sin A + \sin B$$?

Accepted Answer

$$2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$$

Precalculus Trigonometric Identities and Equations – Step-by-Step Study Guidance

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