How do you find the exact values of sin^2(pi/8) using the half angle formula?

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How do you find the exact values of sin^2(pi/8) using the half angle formula?

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Answer 1 · 2015-07-28T15:59:53

I found: $sin (\frac{π}{8}) = 0.3827$ $sin(pi/8)=0.3827$

Explanation:

The half angle formula is:
${sin}^{2} (x) = \frac{1}{2} [1 - cos (2 x)]$ $color(red)(sin^2(x)=1/2[1-cos(2x)]$
if:
$x = \frac{π}{8}$ $x=pi/8$
then $2 x = 2 \frac{π}{8} = \frac{π}{4}$ $2x=2pi/8=pi/4$
You get:
${sin}^{2} (\frac{π}{8}) = \frac{1}{2} [1 - cos (\frac{π}{4})]$ $sin^2(pi/8)=1/2[1-cos(pi/4)]$
but $cos (\frac{π}{4}) = \frac{\sqrt{2}}{2}$ $cos(pi/4)=sqrt(2)/2$
giving:
${sin}^{2} (\frac{π}{8}) = \frac{1}{2} [1 - \frac{\sqrt{2}}{2}] = \frac{2 - \sqrt{2}}{4}$ $sin^2(pi/8)=1/2[1-sqrt(2)/2]=(2-sqrt(2))/4$
and: $sin (\frac{π}{8}) = \pm \sqrt{\frac{2 - \sqrt{2}}{4}} = \pm 0.3827$ $sin(pi/8)=+-sqrt((2-sqrt(2))/4)=+-0.3827$ . I choose the positive one.

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How do you find the exact values of sin^2(pi/8) using the half angle formula?

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