How do you use a half-angle formula to find $cos (\frac{π}{8})$ $cos (pi/8)$ ?

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How do you use a half-angle formula to find $cos (\frac{π}{8})$ $cos (pi/8)$ ?

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Answer 1 · 2015-02-03T09:40:05

The answer is: $\frac{\sqrt{2 + \sqrt{2}}}{2}$ $sqrt(2+sqrt2)/2$ .

The half-angle formula of cosine is:

$cos (\frac{α}{2}) = \sqrt{\frac{1 + cos α}{2}}$ $cos(alpha/2)=sqrt((1+cosalpha)/2)$ .

In this case:

$cos (\frac{π}{8}) = cos (\frac{\frac{π}{4}}{2}) = \sqrt{\frac{1 + cos (\frac{π}{4})}{2}} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{2}}{4}} = \frac{\sqrt{2 + \sqrt{2}}}{2}$ $cos(pi/8)=cos((pi/4)/2)=sqrt((1+cos(pi/4))/2)=sqrt((1+sqrt2/2)/2)=sqrt(((2+sqrt2)/2)/2)=sqrt((2+sqrt2)/4)=sqrt(2+sqrt2)/2$ .

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How do you use a half-angle formula to find $cos (\frac{π}{8})$ $cos (pi/8)$ ?

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