Sine of 36 Degrees

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Theorem

$\sin 36^\circ = \sin \dfrac \pi 5 = \dfrac {\sqrt {\sqrt 5 / \phi} } 2 = \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}$

where $\phi$ denotes the golden mean.


Proof

\(\ds \sin 36^\circ\) \(=\) \(\ds \sqrt {1 - \cos^2 36^\circ}\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \sqrt {1 - \frac {\phi^2} 4}\) Cosine of $36^\circ$
\(\ds \) \(=\) \(\ds \frac 1 2 \sqrt {4 - \phi^2}\)
\(\ds \) \(=\) \(\ds \frac 1 2 \sqrt {4 - \paren {\frac 1 2 + \frac {\sqrt 5} 2}^2}\) Definition 2 of Golden Mean
\(\ds \) \(=\) \(\ds \frac 1 2 \sqrt {4 - \paren {\frac 1 4 + \frac 5 4 + 2 \cdot \frac 1 2 \cdot \frac {\sqrt 5} 2} }\) Square of Sum
\(\ds \) \(=\) \(\ds \frac 1 2 \sqrt {4 - \paren {\frac 3 2 + \frac {\sqrt 5} 2} }\)
\(\ds \) \(=\) \(\ds \frac 1 2 \sqrt {\frac 5 2 - \frac {\sqrt 5} 2}\)
\(\ds \) \(=\) \(\ds \sqrt {\frac 5 8 - \frac {\sqrt 5} 8}\)


We also have:

\(\ds \frac {\sqrt {\sqrt 5 / \phi} } 2\) \(=\) \(\ds \frac 1 2 \sqrt {\sqrt 5 \paren {\phi - 1} }\) Definition 3 of Golden Mean
\(\ds \) \(=\) \(\ds \frac 1 2 \sqrt {\sqrt 5 \paren {\frac 1 2 + \frac {\sqrt 5} 2 - 1} }\) Definition 2 of Golden Mean
\(\ds \) \(=\) \(\ds \frac 1 2 \sqrt {\frac 5 2 - \frac {\sqrt 5} 2}\)
\(\ds \) \(=\) \(\ds \sqrt {\frac 5 8 - \frac {\sqrt 5} 8}\)

$\blacksquare$


Sources

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