Q: Simplify $$\cos(3\theta) \cos(\theta)$$ using a product-to-sum formula.

$$\dfrac{1}{2} [\cos(2\theta) + \cos(4\theta)]$$

Q: Simplify $$\sin(5\theta) + \sin(\theta)$$ using a sum-to-product formula.

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

Q: Simplify $$\cos(7\theta) - \cos(4\theta)$$ using a sum-to-product formula.

$$-2 \sin(\dfrac{11\theta}{2}) \sin(\dfrac{3\theta}{2})$$

Question 1

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

Accepted Answer

$$\sin x \sin y = \dfrac{1}{2} [\cos(x-y) - \cos(x+y)]$$

Question 2

Simplify $$\sin\left(\dfrac{5\pi}{12}\right) \cos\left(\dfrac{\pi}{12}\right)$$ using a product-to-sum formula.

Accepted Answer

$$\dfrac{1}{2} [\sin(\dfrac{6\pi}{12}) + \sin(\dfrac{4\pi}{12})]$$

Question 3

Which of the following is the correct sum-to-product formula for $$\sin x + \sin y$$?

Accepted Answer

$$2 \sin\left(\dfrac{x+y}{2}\right) \cos\left(\dfrac{x-y}{2}\right)$$

Question 4

Simplify $$\cos(3	heta) \cos(	heta)$$ using a product-to-sum formula.

Accepted Answer

$$\dfrac{1}{2} [\cos(2	heta) + \cos(4	heta)]$$

Question 5

Find the exact value of $$\sin\left(\dfrac{5\pi}{3}\right) \cos\left(\dfrac{\pi}{3}\right)$$ using a product-to-sum formula.

Accepted Answer

$$\dfrac{1}{2} [\sin(2\pi) + \sin(\pi)] = 0$$

Question 6

Simplify $$\sin(7x/2) \sin(x/2)$$ using a product-to-sum formula.

Accepted Answer

$$\dfrac{1}{2} [\cos(3x) - \cos(4x)]$$

Question 7

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

Accepted Answer

$$-2 \sin\left(\dfrac{x+y}{2}\right) \sin\left(\dfrac{x-y}{2}\right)$$

Question 8

Simplify $$\sin(5	heta) + \sin(	heta)$$ using a sum-to-product formula.

Accepted Answer

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

Question 9

A sound engineer is analyzing two signals: $$\sin(440t)$$ and $$\sin(442t). $$Using a sum-to-product formula, what is the resulting expression for their sum?

Accepted Answer

$$2 \sin(441t) \cos(t)$$

Question 10

Simplify $$\cos(9x/2) + \cos(7x/2)$$ using a sum-to-product formula.

Accepted Answer

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

Question 11

Find the exact value of $$\sin 75^\circ + \sin 15^\circ$$ using a sum-to-product formula.

Accepted Answer

$$2 \sin 45^\circ \cos 30^\circ = 2 \cdot \dfrac{\sqrt{2}}{2} \cdot \dfrac{\sqrt{3}}{2} = \dfrac{\sqrt{6}}{2}$$

Question 12

Simplify $$\cos(7	heta) - \cos(4	heta)$$ using a sum-to-product formula.

Accepted Answer

$$-2 \sin(\dfrac{11	heta}{2}) \sin(\dfrac{3	heta}{2})$$

Question 13

A physicist is analyzing the interference pattern of two waves represented by $$\sin(\omega t) + \sin((\omega + \Delta \omega)t). $$Which formula best describes the resulting amplitude modulation?

Accepted Answer

$$2 \sin\left(\omega t + \dfrac{\Delta \omega}{2} t\right) \cos\left(\dfrac{\Delta \omega}{2} t\right)$$

Question 14

Simplify $$\sin(3	heta) - \sin(	heta)$$ using a sum-to-product formula.

Accepted Answer

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

Question 15

Find the exact value of $$\sin\left(\dfrac{19\pi}{12}\right) \cos\left(\dfrac{\pi}{12}\right)$$ using a product-to-sum formula.

Accepted Answer

$$\dfrac{1}{2} [\sin(\dfrac{20\pi}{12}) + \sin(\dfrac{18\pi}{12})] = \dfrac{1}{2} [\sin(\dfrac{5\pi}{3}) + \sin(\dfrac{3\pi}{2})] = \dfrac{1}{2} [ -\dfrac{\sqrt{3}}{2} + (-1) ] = -\dfrac{2 + \sqrt{3}}{4}$$

Question 16

Simplify $$\cos(2	heta) + \cos(3	heta)$$ using a sum-to-product formula.

Accepted Answer

$$2 \cos(2.5	heta) \cos(0.5	heta)$$

Question 17

A medical imaging device uses the sum $$\sin(\alpha) + \sin(\beta) to $$model two signals. If $$\alpha = 60^\circ$$ and $$\beta = 120^\circ$$, what is the simplified expression using a sum-to-product formula?

Accepted Answer

$$2 \sin(90^\circ) \cos(30^\circ) = 2 \cdot 1 \cdot \dfrac{\sqrt{3}}{2} = \sqrt{3}$$

Question 18

Simplify $$\sin(5	heta) - \sin(3	heta)$$ using a sum-to-product formula.

Accepted Answer

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

Product-to-Sum and Sum-to-Product Formulas in Trigonometry 5.4

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