Q: A medical imaging device uses the signal $$\cos(2\theta) \cos(8\theta). $$Express this as a sum using a Product-to-Sum formula.

$$\frac{1}{2}[\cos(10\theta) + \cos(6\theta)]$$

Question 1

Which of the following is the correct Product-to-Sum formula for $$\sin A \cos B$$?

Accepted Answer

$$\frac{1}{2}[\sin(A+B) + \sin(A-B)]$$

Question 2

Express $$\sin(70^\circ) \cos(20^\circ) as a $$sum using a Product-to-Sum formula.

Accepted Answer

$$\frac{1}{2}[\sin(90^\circ) + \sin(50^\circ)]$$

Question 3

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

Accepted Answer

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

Question 4

Express $$\cos(80^\circ) \cos(20^\circ) as a $$sum using a Product-to-Sum formula.

Accepted Answer

$$\frac{1}{2}[\cos(100^\circ) + \cos(60^\circ)]$$

Question 5

A sound engineer needs to combine two signals modeled by $$\sin(120t)$$ and $$\sin(100t). $$Using a Sum-to-Product formula, what is $$\sin(120t) + \sin(100t)$$?

Accepted Answer

$$2 \sin(110t) \cos(10t)$$

Question 6

Express $$\sin(75^\circ) \sin(15^\circ) as a $$sum using a Product-to-Sum formula.

Accepted Answer

$$\frac{1}{2}[\cos(60^\circ) - \cos(90^\circ)]$$

Question 7

Which of the following is the correct Product-to-Sum formula for $$\cos A \cos B$$?

Accepted Answer

$$\frac{1}{2}[\cos(A+B) + \cos(A-B)]$$

Question 8

A physicist is analyzing two waveforms: $$\sin(5x)$$ and $$\sin(3x). $$What is $$\sin(5x) - \sin(3x)$$ using a Sum-to-Product formula?

Accepted Answer

$$2 \cos(4x) \sin(x)$$

Question 9

Express $$\cos(120^\circ) \sin(30^\circ) as a $$sum using a Product-to-Sum formula.

Accepted Answer

$$\frac{1}{2}[\sin(150^\circ) - \sin(90^\circ)]$$

Question 10

Which of the following is the correct Product-to-Sum formula for $$\sin A \sin B$$?

Accepted Answer

$$\frac{1}{2}[\cos(A-B) - \cos(A+B)]$$

Question 11

A medical imaging device uses the signal $$\cos(2	heta) \cos(8	heta). $$Express this as a sum using a Product-to-Sum formula.

Accepted Answer

$$\frac{1}{2}[\cos(10	heta) + \cos(6	heta)]$$

Question 12

Express $$\sin(80^\circ) - \sin(20^\circ) as a $$product using a Sum-to-Product formula.

Accepted Answer

$$2 \cos(50^\circ) \sin(30^\circ)$$

Question 13

A chemist is analyzing oscillations modeled by $$\cos(7t) - \cos(3t). $$Express this as a product using a Sum-to-Product formula.

Accepted Answer

$$-2 \sin(5t) \sin(2t)$$

Question 14

Express $$\sin(105^\circ) \sin(15^\circ) as a $$sum using a Product-to-Sum formula.

Accepted Answer

$$\frac{1}{2}[\cos(90^\circ) - \cos(120^\circ)]$$

Question 15

Which of the following is the correct Sum-to-Product formula for $$\cos A - \cos B$$?

Accepted Answer

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

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

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