Sum to Product — Practice Quiz

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

Formulas & key concepts

Sum of Sines

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

Difference of Sines

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

Sum of Cosines

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

Difference of Cosines

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

Practice quiz

  1. Simplify the expression $\sin(5x) + \sin(3x)$.

    • $2\sin(4x)\cos(x)$
    • $2\cos(4x)\sin(x)$
    • $2\sin(8x)\cos(2x)$
    • $2\cos(8x)\sin(2x)$

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

  2. Express $\sin(7\theta) - \sin(3\theta)$ in product form.

    • $2\sin(5\theta)\cos(2\theta)$
    • $2\cos(5\theta)\sin(2\theta)$
    • $2\sin(10\theta)\cos(4\theta)$
    • $2\cos(10\theta)\sin(4\theta)$

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

  3. What is the simplified form of $\cos(4A) + \cos(2A)$?

    • $2\sin(3A)\cos(A)$
    • $2\cos(3A)\sin(A)$
    • $2\cos(3A)\cos(A)$
    • $2\sin(3A)\sin(A)$

    Answer: $2\cos(3A)\cos(A)$

  4. Simplify $\cos(6x) - \cos(2x)$.

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

    Answer: $-2\sin(4x)\sin(2x)$

  5. If $\sin(A) + \sin(B) = 2\sin(40^\circ)\cos(10^\circ)$, what are possible values for $A$ and $B$?

    • $A=50^\circ, B=30^\circ$
    • $A=30^\circ, B=50^\circ$
    • $A=40^\circ, B=10^\circ$
    • $A=50^\circ, B=40^\circ$

    Answer: $A=50^\circ, B=30^\circ$

  6. Which of the following expressions is equivalent to $2\cos(P)\sin(Q)$?

    • $\sin(P+Q) + \sin(P-Q)$
    • $\sin(P+Q) - \sin(P-Q)$
    • $\cos(P+Q) + \cos(P-Q)$
    • $\cos(P+Q) - \cos(P-Q)$

    Answer: $\sin(P+Q) - \sin(P-Q)$

  7. Evaluate $\cos(75^\circ) + \cos(15^\circ)$.

    • $\frac{\sqrt{6}}{2}$
    • $\frac{\sqrt{2}}{2}$
    • $\frac{\sqrt{3}}{2}$
    • $1$

    Answer: $\frac{\sqrt{6}}{2}$

  8. Simplify $\frac{\sin(3x) + \sin(x)}{\cos(3x) + \cos(x)}$.

    • $\tan(x)$
    • $\cot(x)$
    • $\tan(2x)$
    • $\cot(2x)$

    Answer: $\tan(2x)$

  9. Which expression is equivalent to $-2\sin(5x)\sin(x)$?

    • $\cos(6x) - \cos(4x)$
    • $\cos(4x) - \cos(6x)$
    • $\sin(6x) - \sin(4x)$
    • $\sin(4x) - \sin(6x)$

    Answer: $\cos(6x) - \cos(4x)$

  10. The formula $\sin \alpha - \sin \beta = 2\cos(\frac{\alpha + \beta}{2})\sin(\frac{\alpha - \beta}{2})$ is known as the:

    • Sum of Sines identity
    • Difference of Sines identity
    • Sum of Cosines identity
    • Difference of Cosines identity

    Answer: Difference of Sines identity

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