How do you use half angle formulas to find the exact value for cos 15 without using a calculator?

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Answer 1 · 2015-04-23T12:23:38

I consider $cos(30°)=sqrt(3)/2$ as a fundamental value so:
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The last expression can be calculated by hand.

Answer 2 · 2015-04-23T12:35:31

One version of the Half Angle Formula for Cosine is
$cos^2(theta/2) = 1/2(1+cos(theta))$

So
$cos^2(15^o)$

$=cos^2(30^o) = 1/2(1+cos(30^o))$

We know $cos(30^o) = sqrt(3)/2$
(it's one of the standard trig triangles)

So
$cos^2(15^o) = 1/2(1+sqrt(3)/2))$

$cos(15^o) = sqrt(2+sqrt(3))/2$