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

Sep 25, 2015

Find sin (pi/12); cos (pi/12) and tan ((7pi)/12)

#### Explanation:

Call $\sin \frac{\pi}{12} = \sin t .$ Use trig identity: $\cos 2 a = 1 - {\sin}^{2} a$
$\cos 2 \frac{\pi}{12} = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} = 1 - {\sin}^{2} t .$
${\sin}^{2} t = 1 - \frac{\sqrt{3}}{2} = \frac{2 - \sqrt{3}}{4}$
$\sin t = \pm \frac{\sqrt{2 - \sqrt{3}}}{2}$. Since $\frac{\pi}{12}$ has its sin positive, then
$\sin \left(\frac{\pi}{12}\right) = \sin t = \frac{\sqrt{2 - \sqrt{3}}}{2}$

Call $\cos \left(\frac{\pi}{2}\right) = \cos t$
$\cos \left(\frac{2 \pi}{12}\right) = \frac{\sqrt{3}}{2} = 2 {\cos}^{2} t - 1$
${\cos}^{2} t = \frac{2 + \sqrt{3}}{4}$
$\cos t = \pm \frac{\sqrt{2 + \sqrt{3}}}{2}$. Since cos $\left(\frac{\pi}{12}\right)$ is positive, then
$\cos \left(\frac{\pi}{12}\right) = \cos t = \frac{\sqrt{2 + \sqrt{3}}}{2}$

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