Double Angle — Hard Practice Quiz

A Trigonometry cheat sheet for Double Angle — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Sine Double Angle

$$\sin(2\alpha) = 2\sin \alpha \cos \alpha$$

Cosine Double Angle

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

Tangent Double Angle

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

Cotangent Double Angle

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

Practice quiz

  1. Which of the following is the correct expansion of $\sin(4\alpha)$ in terms of $\sin \alpha$ and $\cos \alpha$?

    • $4\sin\alpha\cos\alpha(\cos^2\alpha - \sin^2\alpha)$
    • $2\sin\alpha\cos\alpha(1 - 2\sin^2\alpha)$
    • $4\sin\alpha\cos\alpha(1 - 2\sin^2\alpha)$
    • $2\sin(2\alpha)\cos(2\alpha)$

    Answer: $4\sin\alpha\cos\alpha(\cos^2\alpha - \sin^2\alpha)$

  2. Given $\tan \alpha = x$, which of the following correctly expresses $\cot(2\alpha)$ in terms of $x$?

    • $\frac{1 - x^2}{2x}$
    • $\frac{2x}{1 - x^2}$
    • $\frac{x^2 - 1}{2x}$
    • $\frac{2}{x(1 - x^2)}$

    Answer: $\frac{1 - x^2}{2x}$

  3. If $\cos(2\alpha) = \frac{1}{3}$, what is the value of $\sin^2 \alpha$?

    • $\frac{1}{3}$
    • $\frac{2}{3}$
    • $\frac{1}{6}$
    • $\frac{4}{3}$

    Answer: $\frac{1}{3}$

  4. Given that $\sin \alpha = \frac{3}{5}$ and $\alpha$ is an angle in the first quadrant, determine the value of $\tan(2\alpha)$.

    • $\frac{24}{7}$
    • $\frac{7}{24}$
    • $\frac{12}{5}$
    • $\frac{25}{7}$

    Answer: $\frac{24}{7}$

  5. Simplify the expression $\frac{\sin(2\alpha)}{1 + \cos(2\alpha)}$.

    • $\tan\alpha$
    • $\cot\alpha$
    • $\sin\alpha$
    • $\cos\alpha$

    Answer: $\tan\alpha$

  6. If $\cot \alpha = x$, which of the following expressions represents $\sin(2\alpha)$?

    • $\frac{2x}{x^2 + 1}$
    • $\frac{2}{x^2 + 1}$
    • $\frac{x^2 - 1}{x^2 + 1}$
    • $\frac{2x}{x^2 - 1}$

    Answer: $\frac{2x}{x^2 + 1}$

  7. The expression $\cos^4 \alpha - \sin^4 \alpha$ is equivalent to which of the following?

    • $\cos(2\alpha)$
    • $\sin(2\alpha)$
    • $1$
    • $\cos^2(2\alpha)$

    Answer: $\cos(2\alpha)$

  8. Given that $\sin \alpha + \cos \alpha = k$, express $\sin(2\alpha)$ in terms of $k$.

    • $k^2 - 1$
    • $1 - k^2$
    • $2k$
    • $\frac{k^2 - 1}{2}$

    Answer: $k^2 - 1$

  9. If $\tan(2\alpha) = \frac{4}{3}$ and $\alpha$ is an acute angle, what is the value of $\tan \alpha$?

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

    Answer: $\frac{1}{2}$

  10. Suppose $\cos \alpha = \frac{1}{\sqrt{10}}$ and $\alpha$ is an angle in the fourth quadrant. What is the value of $\sin(2\alpha)$?

    • $-\frac{3}{5}$
    • $\frac{3}{5}$
    • $-\frac{3}{\sqrt{10}}$
    • $\frac{3}{\sqrt{10}}$

    Answer: $-\frac{3}{5}$

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