# A triangle has sides A, B, and C. The angle between sides A and B is (5pi)/12 and the angle between sides B and C is pi/6. If side B has a length of 3, what is the area of the triangle?

Jan 7, 2016

$2.25$

#### Explanation:

Area $= \frac{1}{2} \cdot B \cdot h$ (see sketch)
$B = x + y$
$\tan \left(\frac{5 \pi}{12}\right) = \frac{h}{x}$ and $\tan \left(\frac{\pi}{6}\right) = \frac{h}{y} = \frac{h}{B - x}$
$h = x \tan \left(\frac{5 \pi}{12}\right) = \left(B - x\right) \tan \left(\frac{\pi}{6}\right)$
$\therefore x \tan \left(\frac{5 \pi}{12}\right) + x \tan \left(\frac{\pi}{6}\right) = 3 \tan \left(\frac{\pi}{6}\right)$
$x = \frac{3 \tan \left(\frac{\pi}{6}\right)}{\tan \left(\frac{5 \pi}{12}\right) + \tan \left(\frac{\pi}{6}\right)}$
$\therefore h = \tan \left(\frac{5 \pi}{12}\right) \cdot \frac{3 \tan \left(\frac{\pi}{6}\right)}{\tan \left(\frac{5 \pi}{12}\right) + \tan \left(\frac{\pi}{6}\right)}$
$\approx \frac{3.73 \cdot 3 \cdot 0.577}{3.73 + 0.577}$
$\approx \frac{6.46}{4.31} = 1.5$
Area $= 0.5 \cdot 3 \cdot 1.5 = 2.25$