Sine of 285 Degrees

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Theorem

$\sin 285^\circ = \sin \dfrac {19 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$

where $\sin$ denotes the sine function.


Proof

\(\ds \sin 285^\circ\) \(=\) \(\ds \sin \left({360^\circ - 75^\circ}\right)\)
\(\ds \) \(=\) \(\ds - \sin 75^\circ\) Sine of Conjugate Angle
\(\ds \) \(=\) \(\ds - \dfrac {\sqrt 6 + \sqrt 2} 4\) Sine of $75^\circ$

$\blacksquare$


Sources