Product to Sum — Practice Quiz

A Trigonometry cheat sheet for Product to Sum — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Product of Sines

$$\sin \alpha \cdot \sin \beta = \frac{1}{2}[\cos(\alpha - \beta) - \cos(\alpha + \beta)]$$

Product of Cosines

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

Product of Sine and Cosine

$$\sin \alpha \cdot \cos \beta = \frac{1}{2}[\sin(\alpha + \beta) + \sin(\alpha - \beta)]$$

Practice quiz

  1. Express $2 \sin(3x) \sin(5x)$ as a sum or difference of cosines.

    • $\cos(2x) - \cos(8x)$
    • $\cos(8x) - \cos(2x)$
    • $\sin(8x) - \sin(2x)$
    • $\sin(2x) - \sin(8x)$

    Answer: $\cos(2x) - \cos(8x)$

  2. Simplify the expression $\cos(7x) \cos(2x)$ using a product-to-sum formula.

    • $\frac{1}{2}[\cos(5x) + \cos(9x)]$
    • $\frac{1}{2}[\cos(9x) - \cos(5x)]$
    • $\frac{1}{2}[\sin(9x) + \sin(5x)]$
    • $\frac{1}{2}[\sin(9x) - \sin(5x)]

    Answer: $\frac{1}{2}[\cos(5x) + \cos(9x)]$

  3. Convert $\sin(4\theta) \cos(\theta)$ into a sum or difference of sines.

    • $\frac{1}{2}[\sin(5\theta) + \sin(3\theta)]$
    • $\frac{1}{2}[\sin(5\theta) - \sin(3\theta)]$
    • $\frac{1}{2}[\cos(5\theta) + \cos(3\theta)]$
    • $\frac{1}{2}[\cos(5\theta) - \cos(3\theta)]

    Answer: $\frac{1}{2}[\sin(5\theta) + \sin(3\theta)]$

  4. Which product expression is equivalent to $\frac{1}{2}[\cos(A) - \cos(B)]$?

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

    Answer: $\sin(\frac{A+B}{2}) \sin(\frac{B-A}{2})$

  5. Evaluate $\sin(75^{\circ}) \sin(15^{\circ})$.

    • $\frac{1}{4}$
    • $\frac{\sqrt{3}}{4}$
    • $\frac{1}{2}$
    • $0$

    Answer: $\frac{1}{4}$

  6. Simplify $2 \sin(x) \cos(y) - 2 \cos(x) \sin(y)$.

    • $2 \sin(x-y)$
    • $2 \cos(x-y)$
    • $2 \sin(x+y)$
    • $2 \cos(x+y)$

    Answer: $2 \sin(x-y)$

  7. Which of the following is the correct expansion of $\cos(A) \cos(B)$?

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

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

  8. If $\sin(x) \cos(y) = \frac{1}{4}$ and $\sin(x+y) = \frac{3}{4}$, what is the value of $\sin(x-y)$?

    • $-\frac{1}{4}$
    • $\frac{1}{4}$
    • $\frac{1}{2}$
    • $-\frac{1}{2}$

    Answer: $-\frac{1}{4}$

  9. Simplify $\cos(\frac{5\pi}{12}) \cos(\frac{\pi}{12})$.

    • $\frac{1}{4}$
    • $\frac{\sqrt{3}}{4}$
    • $\frac{1}{2}$
    • $0$

    Answer: $\frac{1}{4}$

  10. Which expression is equivalent to $\sin(A) \sin(B) + \cos(A) \cos(B)$?

    • $\cos(A-B)$
    • $\cos(A+B)$
    • $\sin(A-B)$
    • $\sin(A+B)$

    Answer: $\cos(A-B)$

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