# How do you find the exact values of tan(5pi/12) & sin(5pi/12) using the half angle formula?

Jul 16, 2015

Find $\tan \left(\frac{5 \pi}{12}\right)$ and sin ((5pi)/12)
Answer: $\pm \left(2 \pm \sqrt{3}\right) \mathmr{and} \pm \frac{\sqrt{2 + \sqrt{3}}}{2}$

#### Explanation:

Call tan ((5pi/12) = t.
Use trig identity: $\tan 2 a = \frac{2 \tan a}{1 - {\tan}^{2} a}$
$\tan \left(\frac{10 \pi}{12}\right) = \tan \left(\frac{5 \pi}{6}\right) = - \frac{1}{\sqrt{3}} = \frac{2 t}{1 - {t}^{2}}$
${t}^{2} - 2 \sqrt{3} t - 1 = 0$

$D = {d}^{2} = {b}^{2} - 4 a c = 12 + 4 = 16$--> $d = \pm 4$

$t = \tan \left(\frac{5 \pi}{12}\right) = \frac{2 \sqrt{3}}{2} \pm \frac{4}{2} = 2 \pm \sqrt{3}$

Call $\sin \left(\frac{5 \pi}{12}\right) = \sin y$
Use trig identity: $\cos 2 a = 1 - 2 {\sin}^{2} a$
$\cos \left(\frac{10 \pi}{12}\right) = \cos \left(\frac{5 \pi}{6}\right) = \frac{- \sqrt{3}}{2} = 1 - 2 {\sin}^{2} y$
${\sin}^{2} y = \frac{2 + \sqrt{3}}{4}$
$\sin y = \sin \left(\frac{5 \pi}{12}\right) = \pm \frac{\sqrt{2 + \sqrt{3}}}{2}$