Cosine of 75 Degrees

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Theorem

$\cos 75^\circ = \cos \dfrac {5 \pi}{12} = \dfrac {\sqrt 6 - \sqrt 2} 4$

where $\cos$ denotes the cosine.


Proof

\(\displaystyle \cos 75^\circ\) \(=\) \(\displaystyle \cos \left({90^\circ - 15^\circ}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \sin 15^\circ\) Cosine of Complement equals Sine
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\sqrt 6 - \sqrt 2} 4\) Sine of $15^\circ$

$\blacksquare$


Sources