Powers of Functions — Practice Quiz

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

Formulas & key concepts

Sine Squared

$$\sin^2 \alpha = \frac{1 - \cos(2\alpha)}{2}$$

Cosine Squared

$$\cos^2 \alpha = \frac{1 + \cos(2\alpha)}{2}$$

Sine Cubed

$$\sin^3 \alpha = \frac{3\sin \alpha - \sin(3\alpha)}{4}$$

Cosine Cubed

$$\cos^3 \alpha = \frac{3\cos \alpha + \cos(3\alpha)}{4}$$

Practice quiz

  1. What is the equivalent expression for $\sin^2 \alpha$?

    • $\frac{1 - \cos(2\alpha)}{2}$
    • $\frac{1 + \cos(2\alpha)}{2}$
    • $\frac{3\sin \alpha - \sin(3\alpha)}{4}$
    • $\frac{3\cos \alpha + \cos(3\alpha)}{4}$

    Answer: $\frac{1 - \cos(2\alpha)}{2}$

  2. Which of the following is the correct expansion for $\cos^2 \theta$?

    • $\frac{1 - \cos(2\theta)}{2}$
    • $\frac{1 + \cos(2\theta)}{2}$
    • $\frac{3\sin \theta - \sin(3\theta)}{4}$
    • $\frac{3\cos \theta + \cos(3\theta)}{4}$

    Answer: $\frac{1 + \cos(2\theta)}{2}$

  3. The expression $\sin^3 x$ can be rewritten as:

    • $\frac{1 - \cos(2x)}{2}$
    • $\frac{1 + \cos(2x)}{2}$
    • $\frac{3\sin x - \sin(3x)}{4}$
    • $\frac{3\cos x + \cos(3x)}{4}$

    Answer: $\frac{3\sin x - \sin(3x)}{4}$

  4. Simplify $\cos^3 \beta$ using the power-reduction formula.

    • $\frac{1 - \cos(2\beta)}{2}$
    • $\frac{1 + \cos(2\beta)}{2}$
    • $\frac{3\sin \beta - \sin(3\beta)}{4}$
    • $\frac{3\cos \beta + \cos(3\beta)}{4}$

    Answer: $\frac{3\cos \beta + \cos(3\beta)}{4}$

  5. If $\cos(2x) = 0.5$, what is the value of $\sin^2 x$?

    • $0.25$
    • $0.5$
    • $0.75$
    • $1$

    Answer: $0.25$

  6. Given $\cos(2\theta) = -\frac{1}{2}$, find the value of $\cos^2 \theta$.

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

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

  7. Express $\sin(3\alpha)$ in terms of $\sin \alpha$ and $\sin^3 \alpha$.

    • $3\sin \alpha - 4\sin^3 \alpha$
    • $4\sin^3 \alpha - 3\sin \alpha$
    • $\frac{3\sin \alpha + \sin^3 \alpha}{4}$
    • $\frac{4\sin^3 \alpha + \sin(3\alpha)}{3}$

    Answer: $3\sin \alpha - 4\sin^3 \alpha$

  8. Which expression correctly represents $\cos(3x)$?

    • $3\cos x - 4\cos^3 x$
    • $4\cos^3 x - 3\cos x$
    • $\frac{3\cos x - \cos^3 x}{4}$
    • $\frac{4\cos^3 x + \cos(3x)}{3}$

    Answer: $4\cos^3 x - 3\cos x$

  9. Simplify the expression $\sin^2(3x) + \cos^2(3x)$.

    • $1$
    • $\cos(6x)$
    • $\sin(6x)$
    • $2$

    Answer: $1$

  10. Calculate the exact value of $\sin^3(\frac{\pi}{2})$.

    • $0$
    • $1$
    • $-1$
    • $\frac{1}{2}$

    Answer: $1$

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