# A triangle has sides A, B, and C. The angle between sides A and B is pi/3 and the angle between sides B and C is pi/12. If side B has a length of 42, what is the area of the triangle?

Area$= 204.669$ square units

#### Explanation:

Solve for the altitiude $h$ from angle $B$ to side $b$:

$b = h \cot A + h \cot C$
$b = h \left(\cot A + \cot C\right)$

$h = \frac{b}{\cot A + \cot C} = \frac{42}{\cot \frac{\pi}{12} + \cot \frac{\pi}{3}}$

$h = 9.74613$

Solve the Area:

Area $= \frac{1}{2} \cdot b \cdot h$

Area $= \frac{1}{2} \cdot \left(42\right) \left(9.74613\right)$

Area $= 204.669$ square units