# How do you use the half angle identity to find exact value of Cos(-5pi/12)?

Aug 10, 2015

Find $\cos \left(\frac{- 5 \pi}{12}\right)$

Ans: $+ \frac{\sqrt{\sqrt{3} - 2}}{2}$

#### Explanation:

Call $\cos \left(\frac{- 5 \pi}{12}\right) = \cos t$
$\cos 2 t = \cos \left(\frac{- 10 \pi}{12}\right) = \cos \left(\frac{5 \pi}{6}\right) = - \cos \left(\frac{\pi}{6}\right) = - \frac{\sqrt{3}}{2}$
Use trig identity: $\cos 2 t = 2 {\cos}^{2} t - 1$

$\cos 2 t = \frac{\sqrt{3}}{2} = 2 {\cos}^{2} t - 1.$
$2 {\cos}^{2} t = \frac{\sqrt{3}}{2} - 1 = \frac{\sqrt{3} - 2}{2}$
${\cos}^{2} t = \frac{\sqrt{3} - 2}{4}$

$\cos t = \cos \left(\frac{- 5 \pi}{12}\right) = \pm \frac{\sqrt{\sqrt{3} - 2}}{2}$
Since $\frac{- 5 \pi}{4}$ is in quadrant I, then only the positive answer is accepted -->$\frac{\sqrt{\sqrt{3} - 2}}{2}$
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