# A triangle has sides A, B, and C. The angle between sides A and B is pi/12 and the angle between sides B and C is pi/12. If side B has a length of 3, what is the area of the triangle?

Dec 23, 2015

Area $= \frac{9}{4 \left(2 + \sqrt{3}\right)}$

#### Explanation:

(see diagram)

By Half Angle Formula for $\tan$
$\textcolor{w h i t e}{\text{XXX}} \tan \left(\frac{\pi}{12}\right) = \sin \frac{\frac{\pi}{6}}{1 + \cos \left(\frac{\pi}{6}\right)}$
using standard values for $\sin$ and $\cos$
$\textcolor{w h i t e}{\text{XXX}} \tan \left(\frac{\pi}{12}\right) = \frac{1}{2 + \sqrt{3}}$

The height of the triangle is
$\textcolor{w h i t e}{\text{XXX}} h = \frac{3}{2} \times \tan \left(\frac{\pi}{12}\right)$

and the area is
$\frac{\textcolor{w h i t e}{\text{XXX")("base " xx " height}}}{2}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{3 \times \frac{3}{2} \times \tan \left(\frac{\pi}{2}\right)}{2}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{9}{4} \times \frac{1}{2 + \sqrt{3}}$