# Sine of 165 Degrees

## Theorem

$\sin 165^\circ = \sin \dfrac {11 \pi} {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$

where $\sin$ denotes the sine function.

## Proof

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle \sin 165^\circ$$ $$=$$ $$\displaystyle$$ $$\displaystyle \sin \left({90^\circ + 75^\circ}\right)$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \cos 75^\circ$$ $$\displaystyle$$ $$\displaystyle$$ Sine of Angle plus Right Angle $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \frac {\sqrt 6 - \sqrt 2} 4$$ $$\displaystyle$$ $$\displaystyle$$ Cosine of 75 Degrees

$\blacksquare$