Q: Which identity correctly expresses $$\tan(A-B) in $$terms of $$\tan A$$ and $$\tan B$$?

$$\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$

Q: Which of the following is the double-angle identity for sine?

$$\sin(2\theta) = 2\sin\theta\cos\theta$$

Q: Which of the following is the correct double-angle identity for cosine?

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

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

$$\frac{2 + 3}{1 - 2 \times 3} = -1$$

Question 1

Which of the following is the correct formula for the cosine of the sum of two angles $$A$$ and $$B$$?

Accepted Answer

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

Question 2

Given $$\sin 30^\circ = \frac{1}{2}$$ and $$\cos 60^\circ = \frac{1}{2}$$, what is $$\sin(30^\circ + 60^\circ)$$?

Accepted Answer

$$\frac{\sqrt{3}}{2}$$

Question 3

Which identity correctly expresses $$	an(A-B) in $$terms of $$	an A$$ and $$	an B$$?

Accepted Answer

$$	an(A-B) = \frac{	an A - 	an B}{1 + 	an A 	an B}$$

Question 4

If $$\cos 	heta = \frac{3}{5}$$ and $$	heta is in $$the first quadrant, what is $$\cos(2	heta)$$?

Accepted Answer

$$\frac{7}{25}$$

Question 5

Which of the following is the double-angle identity for sine?

Accepted Answer

$$\sin(2	heta) = 2\sin	heta\cos	heta$$

Question 6

A sound wave can be modeled by $$y = \sin(2x) + \cos(2x). $$Using sum-to-product identities, rewrite this as a single sine function.

Accepted Answer

$$y = \sqrt{2}\sin\left(2x + \frac{\pi}{4}\right)$$

Question 7

Which of the following is a correct application of the cosine of a difference identity?

Accepted Answer

$$\cos(75^\circ) = \cos(120^\circ - 45^\circ) = \cos 120^\circ \cos 45^\circ - \sin 120^\circ \sin 45^\circ$$

Question 8

Solve for $$x in $$the interval $$[0, 2\pi)$$: $$\sin(2x) = 0.$$

Accepted Answer

$$x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$$

Question 9

Which of the following is the correct double-angle identity for cosine?

Accepted Answer

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

Question 10

A Ferris wheel rotates such that the height $$h (in $$meters) of a passenger above the ground at time $$t (in $$seconds) is modeled by $$h = 10 + 8\sin(\omega t). If$$ $$\omega = \frac{\pi}{6}$$, what is the height after $$t = 9$$ seconds?

Accepted Answer

$$h = 10 + 8\sin\left(\frac{3\pi}{2}\right) = 2$$

Question 11

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

Accepted Answer

$$\frac{2 + 3}{1 - 2 	imes 3} = -1$$

Question 12

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

Accepted Answer

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

Question 13

If $$\sin 	heta = \frac{5}{13}$$ and $$	heta is in $$the first quadrant, what is $$\cos(2	heta)$$?

Accepted Answer

$$\cos(2	heta) = 1 - 2\sin^2$	heta = 1 - 2\left(\frac{25}{169}ight) = \frac{119}{169}$$

Question 14

A satellite dish is designed so that the angle between two support beams is $$75^\circ. If $$the beams are positioned at $$A = 45^\circ$$ and $$B = 30^\circ$$, what is $$\sin(A+B)$$?

Accepted Answer

$$\sin(75^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$$

Question 15

Which of the following is a valid step in proving the double-angle identity for cosine?

Accepted Answer

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

Sum, Difference, and Double-Angle Identities in Trigonometry

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