# Find exact value of cos(pi/12)?

Jun 27, 2017

$\cos \left(\frac{\pi}{12}\right) = \frac{\sqrt{3} + 1}{2}$

#### Explanation:

According to the law of fractional angle,
$\cos \theta = 2 {\cos}^{2} \left(\frac{\theta}{2}\right) - 1$
In this form we can write,
$\cos \left(\frac{\pi}{6}\right) = 2 {\cos}^{2} \left(\frac{\pi}{12}\right) - 1$
$\implies \sin \left(\frac{\pi}{2} - \frac{\pi}{6}\right) = 2 {\cos}^{2} \left(\frac{\pi}{12}\right) - 1$
$\sin \left(\frac{\pi}{3}\right) = 2 {\cos}^{2} \left(\frac{\pi}{12}\right) - 1$
$\frac{\sqrt{3}}{2} + 1 = 2 {\cos}^{2} \left(\frac{\pi}{12}\right)$
${\cos}^{2} \left(\frac{\pi}{12}\right) = \frac{2 + \sqrt{3}}{2}$ and

$\cos \left(\frac{\pi}{12}\right) = \sqrt{\frac{2 + \sqrt{3}}{2}}$

= $\sqrt{\frac{4 + 2 \sqrt{3}}{4}}$

= $\sqrt{\frac{{\left(\sqrt{3}\right)}^{2} + 2 \sqrt{3} + {1}^{2}}{4}}$

= $\frac{\sqrt{3} + 1}{2}$