Half Angle — Hard Practice Quiz

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

Practice quiz

1. If $\cos \alpha = \frac{1}{2}$ and $\alpha$ is in the fourth quadrant, what is the value of $\tan(\frac{\alpha}{2})$?

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

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

2. If $\alpha$ is an angle such that $180^\circ < \alpha < 270^\circ$, which of the following statements is true regarding the signs of $\sin(\frac{\alpha}{2})$ and $\cos(\frac{\alpha}{2})$?

   • $\sin(\frac{\alpha}{2}) > 0$ and $\cos(\frac{\alpha}{2}) > 0$
   • $\sin(\frac{\alpha}{2}) > 0$ and $\cos(\frac{\alpha}{2}) < 0$
   • $\sin(\frac{\alpha}{2}) < 0$ and $\cos(\frac{\alpha}{2}) > 0$
   • $\sin(\frac{\alpha}{2}) < 0$ and $\cos(\frac{\alpha}{2}) < 0$

   Answer: $\sin(\frac{\alpha}{2}) > 0$ and $\cos(\frac{\alpha}{2}) < 0$

3. Given $\sin \alpha = \frac{4}{5}$ and $\alpha$ is in the second quadrant, find the exact value of $\cos(\frac{\alpha}{2})$.

   • $\frac{2\sqrt{5}}{5}$
   • $\frac{\sqrt{5}}{5}$
   • $-\frac{\sqrt{5}}{5}$
   • $\frac{3\sqrt{5}}{5}$

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

4. If $\tan \alpha = -\frac{12}{5}$ and $\alpha$ is in the fourth quadrant, what is the value of $\sin(\frac{\alpha}{2})$?

   • $\frac{3\sqrt{13}}{13}$
   • $-\frac{2\sqrt{13}}{13}$
   • $\frac{2\sqrt{13}}{13}$
   • $\frac{\sqrt{13}}{13}$

   Answer: $\frac{2\sqrt{13}}{13}$

5. Which pair of double-angle identities is most directly used to prove that $\tan(\frac{\alpha}{2}) = \frac{\sin \alpha}{1 + \cos \alpha}$?

   • $\sin \alpha = 2\sin(\frac{\alpha}{2})\cos(\frac{\alpha}{2})$ and $\cos \alpha = \cos^2(\frac{\alpha}{2}) - \sin^2(\frac{\alpha}{2})$
   • $\sin \alpha = 2\sin(\frac{\alpha}{2})\cos(\frac{\alpha}{2})$ and $\cos \alpha = 2\cos^2(\frac{\alpha}{2}) - 1$
   • $\sin \alpha = 2\sin(\frac{\alpha}{2})\cos(\frac{\alpha}{2})$ and $\cos \alpha = 1 - 2\sin^2(\frac{\alpha}{2})$
   • $\sin \alpha = \frac{2\tan(\frac{\alpha}{2})}{1 + \tan^2(\frac{\alpha}{2})}$ and $\cos \alpha = \frac{1 - \tan^2(\frac{\alpha}{2})}{1 + \tan^2(\frac{\alpha}{2})}$

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

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

   • $\frac{1}{9}$
   • $\frac{5}{9}$
   • $-\frac{1}{9}$
   • $\frac{7}{9}$

   Answer: $\frac{1}{9}$

7. For which values of $\alpha$ is the expression $\frac{1 - \cos \alpha}{\sin \alpha}$ undefined?

   • $\alpha = n\pi$, where $n$ is an integer.
   • $\alpha = \frac{n\pi}{2}$, where $n$ is an integer.
   • $\alpha = 2n\pi$, where $n$ is an integer.
   • $\alpha = (2n+1)\frac{\pi}{2}$, where $n$ is an integer.

   Answer: $\alpha = n\pi$, where $n$ is an integer.

8. Simplify the expression $\frac{\sin(\frac{\alpha}{2}) \cos(\frac{\alpha}{2})}{\tan(\frac{\alpha}{2})}$.

   • $\frac{1 - \cos \alpha}{2}$
   • $\frac{1 + \cos \alpha}{2}$
   • $\sin \alpha$
   • $\cos \alpha$

   Answer: $\frac{1 + \cos \alpha}{2}$

9. If $\tan(\frac{\alpha}{2}) = 3$, find the value of $\cos \alpha$.

   • $\frac{3}{5}$
   • $-\frac{3}{5}$
   • $\frac{4}{5}$
   • $-\frac{4}{5}$

   Answer: $-\frac{4}{5}$

10. Find the exact value of $\sin(15^\circ)$.

    • $\frac{\sqrt{6} + \sqrt{2}}{4}$
    • $\frac{\sqrt{6} - \sqrt{2}}{4}$
    • $\frac{\sqrt{2} - \sqrt{6}}{4}$
    • $\frac{\sqrt{3}}{2}$

    Answer: $\frac{\sqrt{6} - \sqrt{2}}{4}$