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#