Sine of 22.5 Degrees

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Theorem

$\sin 22.5 \degrees = \sin \dfrac \pi 8 = \dfrac 1 2 \sqrt {2 - \sqrt 2}$

where $\sin$ denotes sine.


Proof

\(\ds \sin 22.5 \degrees\) \(=\) \(\ds \sin \frac {45 \degrees} 2\)
\(\ds \) \(=\) \(\ds +\sqrt {\frac {1 - \cos 45 \degrees} 2}\) Half Angle Formula for Sine
\(\ds \) \(=\) \(\ds \sqrt {\frac {1 - \frac {\sqrt 2} 2} 2}\) Cosine of $45 \degrees$
\(\ds \) \(=\) \(\ds \sqrt {\frac {2 - \sqrt 2} 4}\) multiplying top and bottom by 2
\(\ds \) \(=\) \(\ds \frac 1 2 \sqrt {2 - \sqrt 2}\) factoring $\dfrac 1 2$ out of the square root

$\blacksquare$


Also see