# How do you use half angle formula to find sin 67.5?

May 4, 2018

$\frac{\sqrt{2 + \sqrt{2}}}{2}$

#### Explanation:

$\sin {67.5}^{\circ}$

The half angle formula for $\sin \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$.

The $\pm$ sign depends on which quadrant your angle is in. Since ${67.5}^{\circ}$ is in the 1st quadrant and sine is positive in that quadrant, we know that it is positive.

${67.5}^{\circ} \cdot 2 = {135}^{\circ}$, so:
$\sin \left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos {135}^{\circ}}{2}}$

  $\quad \quad \quad \quad \quad \quad \quad = \sqrt{\frac{1 - \left(- \frac{\sqrt{2}}{2}\right)}{2}}$

  $\quad \quad \quad \quad \quad \quad \quad = \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}}$

  $\quad \quad \quad \quad \quad \quad \quad = \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}}$

  $\quad \quad \quad \quad \quad \quad \quad = \sqrt{\frac{2 + \sqrt{2}}{4}}$

  $\quad \quad \quad \quad \quad \quad \quad = \frac{\sqrt{2 + \sqrt{2}}}{2}$

Hope this helps!