Q: Evaluate $$\tan 75^\circ$$ using the tangent sum identity.

$$\tan 75^\circ = \tan(45^\circ + 30^\circ) = \dfrac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ} = 2 + \sqrt{3}$$

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

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

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

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

Q: Which of the following is the correct tangent sum identity?

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

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

Find the exact value of $$\cos 75^\circ$$ using a sum or difference identity.

Accepted Answer

$$\cos 75^\circ = \cos(45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ = \dfrac{\sqrt{6} - \sqrt{2}}{4}$$

Question 3

Which identity correctly expresses $$\sin(A - B)$$?

Accepted Answer

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

Question 4

Evaluate $$	an 75^\circ$$ using the tangent sum identity.

Accepted Answer

$$	an 75^\circ = 	an(45^\circ + 30^\circ) = \dfrac{	an 45^\circ + 	an 30^\circ}{1 - 	an 45^\circ 	an 30^\circ} = 2 + \sqrt{3}$$

Question 5

Which of the following is the double-angle identity for $$\sin 2	heta$$?

Accepted Answer

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

Question 6

A sound wave can be modeled by $$y = \sin(2	heta). $$What is the amplitude of this wave?

Accepted Answer

1

Question 7

Which of the following is the correct double-angle identity for $$\cos 2	heta$$?

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

Which of the following is a correct application of the cosine difference identity to $$\cos(60^\circ - 45^\circ)$$?

Accepted Answer

$$\cos(60^\circ - 45^\circ) = \cos 60^\circ \cos 45^\circ - \sin 60^\circ \sin 45^\circ$$

Question 10

A Ferris wheel rotates such that the height $$h (in $$meters) of a seat above the ground at time $$t (in $$seconds) is modeled by $$h = 10 + 8\sin(\omega t). If$$ $$\omega = 2$$, what is the maximum height reached by the seat?

Accepted Answer

26 meters

Question 11

Which of the following is the correct tangent sum identity?

Accepted Answer

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

Question 12

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

Accepted Answer

$$\cos(2\alpha) = 1 - 2\left(\dfrac{3}{5}\$$right)^2$$ = \dfrac{7}{25}$$

Question 13

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

Accepted Answer

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

Question 14

A satellite dish is designed so that the angle between two support beams is $$	heta. If $$the beams form an angle of $$30^\circ$$ with the ground and $$	heta = 60^\circ$$, what is $$\sin(2	heta)$$?

Accepted Answer

$$\sin(2	heta) = \sin(120^\circ) = \dfrac{\sqrt{3}}{2}$$

Question 15

Which of the following is the correct formula for $$	an(2	heta)$$?

Accepted Answer

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

Question 16

If $$\cos \alpha = \dfrac{4}{5}$$ and $$\alpha is in $$the first quadrant, what is $$\sin(2\alpha)$$?

Accepted Answer

$$\sin(2\alpha) = 2 \sin \alpha \cos \alpha = 2 \cdot \dfrac{3}{5} \cdot \dfrac{4}{5} = \dfrac{24}{25}$$

Sum, Difference, and Double-Angle Trigonometric Identities

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