Sine of 165 Degrees

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Theorem

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

where $\sin$ denotes the sine function.


Proof

\(\ds \sin 165 \degrees\) \(=\) \(\ds \map \sin {90 \degrees + 75 \degrees}\)
\(\ds \) \(=\) \(\ds \cos 75 \degrees\) Sine of Angle plus Right Angle
\(\ds \) \(=\) \(\ds \frac {\sqrt 6 - \sqrt 2} 4\) Cosine of $75 \degrees$

$\blacksquare$


Sources