Sine of 75 Degrees

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Theorem

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

where $\sin$ denotes the sine function.


Proof 1

\(\displaystyle \sin 75^\circ\) \(=\) \(\displaystyle \sin \left({60^\circ + 15^\circ}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \sin 60^\circ \cos 15^\circ + \cos 60^\circ \sin 15^\circ\) Sine of Sum
\(\displaystyle \) \(=\) \(\displaystyle \left({\frac {\sqrt 3} 2}\right) \left({\frac {\sqrt 6 + \sqrt 2} 4}\right) + \left({\frac 1 2}\right) \left({\dfrac {\sqrt 6 - \sqrt 2} 4}\right)\) Sine of $60^\circ$, Cosine of $15^\circ$, Cosine of $60^\circ$, Sine of $15^\circ$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 8 \left({\sqrt 3 \left({\sqrt 6 + \sqrt 2}\right) + \sqrt 6 - \sqrt 2}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 8 \left({\sqrt 3 \sqrt 3 \sqrt 2 + \sqrt 3 \sqrt 2 + \sqrt 3 \sqrt 2 - \sqrt 2}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 8 \left({3 \sqrt 2 + 2 \sqrt 3 \sqrt 2 - \sqrt 2}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 8 \left({2 \sqrt 2 + 2 \sqrt 6}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sqrt 6 + \sqrt 2} 4\)

$\blacksquare$


Proof 2

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

$\blacksquare$


Sources