Sine of 165 Degrees

From ProofWiki
Jump to: navigation, search

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          

$\blacksquare$


Sources