Cosine of 22.5 Degrees
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Theorem
- $\cos 22.5 \degrees = \cos \dfrac \pi 8 = \dfrac 1 2 \sqrt {2 + \sqrt 2}$
where $\cos$ denotes cosine.
Proof
| \(\ds \cos 22.5 \degrees\) | \(=\) | \(\ds \cos \frac {45 \degrees} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds +\sqrt {\frac {1 + \cos 45 \degrees} 2}\) | Half Angle Formula for Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {\frac {1 + \frac {\sqrt 2} 2} 2}\) | 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 |
$\blacksquare$