Half Angle — Practice Quiz

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

Formulas & key concepts

Sine Half Angle

$$\sin\left(\frac{\alpha}{2}\right) = \pm\sqrt{\frac{1 - \cos \alpha}{2}}$$

Cosine Half Angle

$$\cos\left(\frac{\alpha}{2}\right) = \pm\sqrt{\frac{1 + \cos \alpha}{2}}$$

Tangent Half Angle

$$\tan\left(\frac{\alpha}{2}\right) = \frac{\sin \alpha}{1 + \cos \alpha} = \frac{1 - \cos \alpha}{\sin \alpha}$$

Practice quiz

  1. What is the exact value of $\sin(15^\circ)$ using the half-angle formula?

    • $\frac{\sqrt{2 - \sqrt{3}}}{2}$
    • $\frac{\sqrt{2 + \sqrt{3}}}{2}$
    • $\frac{\sqrt{3}}{2}$
    • $\frac{1}{2}$

    Answer: $\frac{\sqrt{2 - \sqrt{3}}}{2}$

  2. If $\cos \alpha = \frac{3}{5}$ and $\alpha$ is in the fourth quadrant, what is $\cos(\frac{\alpha}{2})$?

    • $\frac{2\sqrt{5}}{5}$
    • -$ \frac{2\sqrt{5}}{5}$
    • $\frac{\sqrt{5}}{5}$
    • -$ \frac{\sqrt{5}}{5}$

    Answer: -$ \frac{2\sqrt{5}}{5}$

  3. Which of the following expressions is equivalent to $\tan(\frac{\theta}{2})$?

    • $\frac{\sin \theta}{1 + \cos \theta}$
    • $\frac{1 + \cos \theta}{\sin \theta}$
    • $\frac{1 - \cos \theta}{1 + \sin \theta}$
    • $\frac{\sin \theta}{1 - \cos \theta}$

    Answer: $\frac{\sin \theta}{1 + \cos \theta}$

  4. Given $\sin \alpha = \frac{4}{5}$ and $\alpha$ is in the second quadrant, find $\tan(\frac{\alpha}{2})$?

    • $2$
    • $\frac{1}{2}$
    • -$ \frac{1}{2}$
    • -$ \frac{2}{5}$

    Answer: $2$

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

    • $\frac{7}{9}$
    • $\frac{8}{9}$
    • $\frac{1}{9}$
    • -$ \frac{7}{9}$

    Answer: $\frac{7}{9}$

  6. Which of the following expressions is equivalent to $\frac{\sin(2\theta)}{1 + \cos(2\theta)}$?

    • $\tan(\theta)$
    • $\tan(2\theta)$
    • $\sin(\theta)$
    • $\cos(\theta)$

    Answer: $\tan(\theta)$

  7. If $\cos \theta = -\frac{1}{2}$ and $\theta$ is in the third quadrant, what is $\sin(\frac{\theta}{2})$?

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

    Answer: $\frac{\sqrt{3}}{2}$

  8. Simplify the expression $\frac{1 - \cos(4x)}{\sin(4x)}$.

    • $\tan(2x)$
    • $\tan(4x)$
    • $\cot(2x)$
    • $\sin(2x)$

    Answer: $\tan(2x)$

  9. Given $\cos \beta = 0.6$ and $\beta$ is an acute angle, find $\tan(\frac{\beta}{2})$?

    • $0.5$
    • $2$
    • $0.8$
    • $0.6$

    Answer: $0.5$

  10. Which of the following is the correct half-angle formula for $\cos(\frac{x}{2})$?

    • $\pm\sqrt{\frac{1 + \cos x}{2}}$
    • $\pm\sqrt{\frac{1 - \cos x}{2}}$
    • $\frac{\sin x}{1 + \cos x}$
    • $\frac{1 - \cos x}{\sin x}$

    Answer: $\pm\sqrt{\frac{1 + \cos x}{2}}$

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