Sine of 195 Degrees

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Theorem

$\sin 195 \degrees = \sin \dfrac {13 \pi} {12} = -\dfrac {\sqrt 6 - \sqrt 2} 4$

where $\sin$ denotes the sine function.


Proof

\(\ds \sin 195 \degrees\) \(=\) \(\ds \map \sin {360 \degrees - 165 \degrees}\)
\(\ds \) \(=\) \(\ds -\sin 165 \degrees\) Sine of Conjugate Angle
\(\ds \) \(=\) \(\ds -\frac {\sqrt 6 - \sqrt 2} 4\) Sine of $165 \degrees$

$\blacksquare$


Sources