# Find the exact value of sin(pi/12) , cos (11pi/12) , and tan (7pi/12)?

Aug 19, 2015

$\sin \left(\frac{\pi}{12}\right) = \left(\frac{\sqrt{3} - 1}{2 \sqrt{2}}\right)$
$\cos \left(\frac{11 \pi}{12}\right) = - \left(\frac{1 + \sqrt{3}}{2 \sqrt{2}}\right)$
$\tan \left(\frac{7 \pi}{12}\right) = - 2 - \sqrt{3}$

#### Explanation:

$\sin \left(\frac{\pi}{12}\right)$
$= \sin \left(\frac{\pi}{3} - \frac{\pi}{4}\right)$
$= \sin \left(\frac{\pi}{3}\right) \cos \left(\frac{\pi}{4}\right) - \cos \left(\frac{\pi}{3}\right) \sin \left(\frac{\pi}{4}\right)$
$= \left(\frac{\sqrt{3}}{2}\right) \left(\frac{1}{\sqrt{2}}\right) - \left(\frac{1}{2}\right) \left(\frac{1}{\sqrt{2}}\right)$
$= \left(\frac{\sqrt{3} - 1}{2 \sqrt{2}}\right)$

$\cos \left(\frac{11 \pi}{12}\right)$
$= \cos \left(\frac{2 \pi}{3} + \frac{\pi}{4}\right)$
$= \cos \left(\frac{2 \pi}{3}\right) \cos \left(\frac{\pi}{4}\right) - \sin \left(\frac{2 \pi}{3}\right) \sin \left(\frac{\pi}{4}\right)$
$= \cos \left(\pi - \frac{\pi}{3}\right) \cos \left(\frac{\pi}{4}\right) - \sin \left(\pi - \frac{\pi}{3}\right) \sin \left(\frac{\pi}{4}\right)$
$= - \cos \left(\frac{\pi}{3}\right) \cos \left(\frac{\pi}{4}\right) - \sin \left(\frac{\pi}{3}\right) \sin \left(\frac{\pi}{4}\right)$
$= - \left(\frac{1}{2}\right) \left(\frac{1}{\sqrt{2}}\right) - \left(\frac{\sqrt{3}}{2}\right) \left(\frac{1}{\sqrt{2}}\right)$
$= - \left(\frac{1 + \sqrt{3}}{2 \sqrt{2}}\right)$

$\tan \left(\frac{7 \pi}{12}\right)$
$= \tan \left(\frac{\pi}{3} + \frac{\pi}{4}\right)$
$= \frac{\sqrt{3} + 1}{1 - \sqrt{3}}$
$= \frac{\left(\sqrt{3} + 1\right) \left(1 + \sqrt{3}\right)}{1 - 3}$
$= - 2 - \sqrt{3}$