Use a double or half angle formula to determine value of cos(13π/12)?

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Use a double or half angle formula to determine value of cos(13π/12)?

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Answer 1 · 2018-05-05T00:01:32

$- \frac{2 + \sqrt{3}}{2}$ $- (2 + sqrt3)/2$

Explanation:

$cos (\frac{13 π}{12}) = cos (\frac{π}{12} + π) = - cos (\frac{π}{12})$ $cos ((13pi)/12) = cos (pi/12 + pi) = - cos (pi/12)$
Find $cos (\frac{π}{12})$ $cos ((pi)/12)$ by using trig identity:
$2 {cos}^{2} a = 1 + cos 2 a$ $2cos^2 a = 1 + cos 2a$ .
In this case:
$2 {cos}^{2} (\frac{π}{12}) = 1 + cos (\frac{π}{6}) = 1 + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2}$ $2cos^2 (pi/12) = 1 + cos (pi/6) = 1 + sqrt3/2 = (2 + sqrt3)/2$
${cos}^{2} (\frac{π}{12}) = \frac{2 + \sqrt{3}}{4}$ $cos^2 (pi/12) = (2 + sqrt3)/4$
$cos (\frac{π}{12}) = \frac{2 + \sqrt{3}}{2}$ $cos (pi/12) = (2 + sqrt3)/2$ --> ( $cos (\frac{π}{12})$ $cos (pi/12)$ is positive).
Finally,
$cos (\frac{13 π}{12}) = - cos (\frac{π}{12}) = - \frac{2 + \sqrt{3}}{2}$ $cos ((13pi)/12) = - cos (pi/12) = - (2 + sqrt3)/2$

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Use a double or half angle formula to determine value of cos(13π/12)?

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