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

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Answer 1 · 2015-08-06T01:45:23

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

$cos^2(x) = (1+cos(2x))/2$

By inference:
$cos^2(x/2) = (1+cosx)/2$

Square root to get:

$cos(x/2) = pmsqrt((1+cosx)/2)$
$+$ if quadrant I or IV
$-$ if quadrant II or III

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

$cos(45^o/2) = sqrt((1+cos45^o)/2)$

$= sqrt((1+(sqrt2/2))/2)$

$= sqrt((((2+sqrt2)/2))/2)$

$= sqrt((2+sqrt2)/4)$

$= color(blue)(sqrt(2+sqrt2)/2)$

or $~~0.9238795$