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

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Aug 30, 2015

$\textcolor{red}{\cos 15 = \frac{\sqrt{2 + \sqrt{3}}}{2}}$

Explanation:

The cosine half-angle formula is

cos(x/2) = ±sqrt((1 + cos x) / 2)

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

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

$15 = \frac{30}{2}$

$\cos 15 = \cos \left(\frac{30}{2}\right) = \sqrt{\frac{1 + \cos 30}{2}}$

$\cos 15 = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}}$

$\cos 15 = \frac{\sqrt{2 + \sqrt{3}}}{2}$

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