Sine of 105 Degrees

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Theorem

$\sin 105^\circ = \sin \dfrac {7 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$

where $\sin$ denotes the sine function.


Proof

\(\displaystyle \sin 105^\circ\) \(=\) \(\displaystyle \sin \left({90^\circ + 15^\circ}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \cos 15^\circ\) Sine of Angle plus Right Angle
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sqrt 6 + \sqrt 2} 4\) Cosine of 15 Degrees

$\blacksquare$


Sources