A triangle has sides A,B, and C. If the angle between sides A and B is (7pi)/12, the angle between sides B and C is pi/6, and the length of B is 5, what is the area of the triangle?

Area=$8.53766$ square units

Explanation:

From the given, two angles $A = \frac{\pi}{6}$, $C = \frac{7 \pi}{12}$ and included side $b = 5$. Try drawing the triangle. See that angle $B = \frac{\pi}{4}$ by computation using the formula $A + B + C = \pi$. Also , the altitude from angle C to side c can be called height $h$ is $h = b \cdot \sin \left(\frac{\pi}{6}\right) = 2.5$
Side $c$ can be computed using formula $c = b \cdot \cos A + h \cdot \cot B$.
$c = 5 \cdot \cos \left(\frac{\pi}{6}\right) + 2.5 \cdot \cot \left(\frac{\pi}{4}\right)$=$2.5 \cdot \left(\sqrt{3} + 1\right)$

$c = 2.5 \left(\sqrt{3} + 1\right)$
Area can now be computed

Area$= \frac{1}{2} \cdot b \cdot c \cdot \sin A$

Area$= \frac{1}{2} \cdot 5 \cdot \left(2.5 \left(\sqrt{3} + 1\right)\right) \cdot \sin \left(\frac{\pi}{6}\right)$

Area$= 8.53766$ square units