Q: Express $$\cos(4\theta) \cos(6\theta) as a $$sum containing only cosines.

$$\frac{1}{2} \left[ \cos(10\theta) + \cos(2\theta) \right]$$

Q: Express $$\sin(6\theta) - \sin(4\theta) as a $$product.

$$2 \sin(\theta) \cos(5\theta)$$

Q: Express $$\sin \theta + \sin 3\theta as a $$product.

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

Q: Express $$\cos \left( \frac{5\theta}{2} \right) - \cos \left( \frac{7\theta}{2} \right) as a $$product.

$$-2 \sin(3\theta) \sin(\theta)$$

Question 1

Which of the following is the correct product-to-sum formula for $$\sin \alpha \sin \beta$$?

Accepted Answer

$$\frac{1}{2} \left[ \cos(\alpha - \beta) - \cos(\alpha + \beta) \right]$$

Question 2

Express $$\cos(4	heta) \cos(6	heta) as a $$sum containing only cosines.

Accepted Answer

$$\frac{1}{2} \left[ \cos(10	heta) + \cos(2	heta) ight]$$

Question 3

Which sum-to-product formula correctly expresses $$\sin \alpha + \sin \beta$$?

Accepted Answer

$$2 \sin \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right)$$

Question 4

Express $$\sin(6	heta) - \sin(4	heta) as a $$product.

Accepted Answer

$$2 \sin(	heta) \cos(5	heta)$$

Question 5

Find the exact value of $$\sin(165^\circ) \sin(45^\circ).$$

Accepted Answer

$$\frac{1}{2} \left[ \cos(120^\circ) - \cos(210^\circ) \right]$$

Question 6

Express $$\sin 	heta + \sin 3	heta as a $$product.

Accepted Answer

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

Question 7

Which of the following is the correct sum-to-product formula for $$\cos \alpha - \cos \beta$$?

Accepted Answer

$$-2 \sin \left( \frac{\alpha + \beta}{2} \right) \sin \left( \frac{\alpha - \beta}{2} \right)$$

Question 8

Express $$\cos(225^\circ) - \cos(195^\circ) as a $$product.

Accepted Answer

$$-2 \sin(210^\circ) \sin(15^\circ)$$

Question 9

Which formula expresses $$\sin \alpha \cos \beta as a $$sum?

Accepted Answer

$$\frac{1}{2} \left[ \sin(\alpha + \beta) + \sin(\alpha - \beta) \right]$$

Question 10

Express $$\cos \left( \frac{5	heta}{2} ight) - \cos \left( \frac{7	heta}{2} ight) as a $$product.

Accepted Answer

$$-2 \sin(3	heta) \sin(	heta)$$

Question 11

A sound engineer is analyzing two signals with frequencies $$\alpha$$ and $$\beta. $$Which formula allows her to write $$\sin \alpha \sin \beta as a $$sum of cosines to simplify the analysis?

Accepted Answer

$$\frac{1}{2} \left[ \cos(\alpha - \beta) - \cos(\alpha + \beta) \right]$$

Question 12

Express $$\sin \left( \frac{15	heta}{2} ight) \cos \left( \frac{2	heta}{2} ight) as a $$sum.

Accepted Answer

$$\frac{1}{2} \left[ \sin(\frac{15	heta}{2} + \frac{2	heta}{2}) + \sin(\frac{15	heta}{2} - \frac{2	heta}{2}) ight]$$

Question 13

Which of the following is the correct product-to-sum formula for $$\cos \alpha \cos \beta$$?

Accepted Answer

$$\frac{1}{2} \left[ \cos(\alpha - \beta) + \cos(\alpha + \beta) \right]$$

Question 14

Express $$\sin(4	heta) + \sin(2	heta) as a $$product.

Accepted Answer

$$2 \sin(3	heta) \cos(	heta)$$

Question 15

A physicist is modeling a wave using $$\sin(\omega t) \cos(\omega t). $$Which formula can be used to rewrite this as a sum?

Accepted Answer

$$\frac{1}{2} \left[ \sin(2\omega t) \right]$$

Question 16

Express $$\cos(\alpha) + \cos(\beta) as a $$product.

Accepted Answer

$$2 \cos \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right)$$

Product-to-Sum and Sum-to-Product Formulas in Trigonometry

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