# How do you find the exact values of cos 22.5 degrees using the half angle formula?

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31
Aug 6, 2015

The half angle identity for cosine can be derived (since I don't recall it off-hand):

${\cos}^{2} \left(x\right) = \frac{1 + \cos \left(2 x\right)}{2}$

By inference:
${\cos}^{2} \left(\frac{x}{2}\right) = \frac{1 + \cos x}{2}$

Square root to get:

$\cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}}$
$+$ if quadrant I or IV
$-$ if quadrant II or III

${22.5}^{o}$ is quadrant I, so it is positive.

$\cos \left({45}^{o} / 2\right) = \sqrt{\frac{1 + \cos {45}^{o}}{2}}$

$= \sqrt{\frac{1 + \left(\frac{\sqrt{2}}{2}\right)}{2}}$

$= \sqrt{\frac{\left(\frac{2 + \sqrt{2}}{2}\right)}{2}}$

$= \sqrt{\frac{2 + \sqrt{2}}{4}}$

$= \textcolor{b l u e}{\frac{\sqrt{2 + \sqrt{2}}}{2}}$

or $\approx 0.9238795$

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