Q: Which of the following is the correct formula for the tangent of a sum of two angles $$\alpha$$ and $$\beta$$?

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

Q: Find the exact value of $$\tan 75^\circ$$ using the tangent sum identity.

$$\tan 75^\circ = \tan(45^\circ + 30^\circ) = \frac{\tan 45^\circ + \tan 30^\circ}{1 + \tan 45^\circ \tan 30^\circ}$$, and the exact value is $$2 + \sqrt{3}$$

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

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

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

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

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

$$\cos(2\theta) = 2 \left(\frac{4}{5}\$$right)^2$$ - 1 = \frac{7}{25}$$

Q: Which of the following is the correct formula for $$\tan(\alpha - \beta)$$?

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

Question 1

Which of the following is the correct formula for the cosine of a sum of two angles $$\alpha$$ and $$\beta$$?

Accepted Answer

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

Question 2

What is the exact value of $$\cos 75^\circ$$ using the cosine-of-a-difference identity?

Accepted Answer

$$\cos 75^\circ = \cos(45^\circ + 30^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{1}{2} \cdot \frac{1}{2}$$, and the exact value is $$\frac{\sqrt{6} - \sqrt{2}}{4}$$

Question 3

Which identity correctly expresses $$\sin(\alpha - \beta)$$?

Accepted Answer

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

Question 4

Which of the following is the correct formula for the tangent of a sum of two angles $$\alpha$$ and $$\beta$$?

Accepted Answer

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

Question 5

Find the exact value of $$	an 75^\circ$$ using the tangent sum identity.

Accepted Answer

$$	an 75^\circ = 	an(45^\circ + 30^\circ) = \frac{	an 45^\circ + 	an 30^\circ}{1 + 	an 45^\circ 	an 30^\circ}$$, and the exact value is $$2 + \sqrt{3}$$

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

$$\cos(2	heta) = 2 \left(\frac{4}{5}\$$right)^2$$ - 1 = \frac{7}{25}$$

Question 9

Solve $$\sin(2x) = 0 in $$the interval $$[0, 2\pi]. $$How many solutions are there?

Accepted Answer

There are 4 solutions: $$x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$$

Question 10

A sound wave can be modeled by $$y = \sin(2x). $$How many times does the wave cross the x-axis in the interval $$[0, 2\pi]$$?

Accepted Answer

4 times

Question 11

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

Accepted Answer

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

Question 12

If $$\cos \alpha = \frac{1}{2}$$ and $$\sin \beta = \frac{\sqrt{3}}{2}$$, what is $$\sin(\alpha + \beta)$$?

Accepted Answer

$$\sin(\alpha + \beta) = \sqrt{1 - \left(\frac{1}{2}\$$right)^2$$} \cdot \frac{\sqrt{3}}{2} + \frac{1}{2} \cdot \sqrt{1 - \left(\frac{\sqrt{3}}{2}\$$right)^2$$}$$

Question 13

Which of the following is NOT a valid double-angle identity?

Accepted Answer

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

Question 14

A medical imaging device uses the function $$y = \cos(2x) to $$model signal intensity. What is the period of this function?

Accepted Answer

$$\pi$$

Question 15

If $$\sin \alpha = \frac{1}{3}$$ and $$\cos \beta = \frac{2}{3}$$, what is $$\cos(\alpha + \beta)$$?

Accepted Answer

$$\cos(\alpha + \beta) = \sqrt{1 - \left(\frac{1}{3}\$$right)^2$$} \cdot \frac{2}{3} - \frac{1}{3} \cdot \sqrt{1 - \left(\frac{2}{3}\$$right)^2$$}$$

Question 16

Which of the following best describes the algebraic difference between function identities and real number algebra?

Accepted Answer

Function identities can express relationships between angles and their trigonometric values, while real number algebra is limited to arithmetic operations.

Sum, Difference, and Double-Angle Trigonometric Identities

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