How do you find the exact values of cos 15 using the half angle formula?

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How do you find the exact values of cos 15 using the half angle formula?

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Answer 1 · 2015-08-30T18:36:37

$cos 15 = \frac{\sqrt{2 + \sqrt{3}}}{2}$ $color(red)(cos15 = sqrt(2 +sqrt3)/2)$

Explanation:

The cosine half-angle formula is

$cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + cos x}{2}}$ $cos(x/2) = ±sqrt((1 + cos x) / 2)$

The sign is positive if $\frac{x}{2}$ $x/2$ is in the first or fourth quadrant and negative if $\frac{x}{2}$ $x/2$ is in the second or third quadrant.

$15°$ is in the first quadrant, so the sign is positive.

$15= 30/2$

∴ $cos15 = cos(30/2) = sqrt((1+cos30)/2)$

$cos15 = sqrt((1 + sqrt3/2)/2) = sqrt((2 + sqrt3)/4)$

$cos15 = sqrt(2 +sqrt3)/2$

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How do you find the exact values of cos 15 using the half angle formula?

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