# A triangle has sides A,B, and C. If the angle between sides A and B is (3pi)/4, the angle between sides B and C is pi/6, and the length of B is 7, what is the area of the triangle?

Apr 7, 2018

color(blue)("Area of Triangle " A_t = color(indigo)(33.46 " sq units"

#### Explanation:

$\hat{A} = \frac{\pi}{6} , \hat{C} = \frac{3 \pi}{4} , b = 7 , \text{ To find area of } \Delta$

$\hat{B} = \pi - \frac{\pi}{6} - \frac{3 \pi}{4} = \frac{\pi}{12}$

Applying the Law of Sines,

$\frac{a}{\sin} \left(\frac{\pi}{6}\right) = \frac{7}{\sin} \left(\frac{\pi}{12}\right) = \frac{c}{\sin} \left(\frac{3 \pi}{4}\right)$

$a = \frac{7 \cdot \sin \left(\frac{\pi}{6}\right)}{\sin} \left(\frac{\pi}{12}\right) = 13.52$

Now we know two sides and the included angle.

$\therefore \text{Area of } \Delta = {A}_{t} = \left(\frac{1}{2}\right) a b \sin C$

A_t = (1/2) * 13.52 * 7 * sin ((3pi)/4) = color(indigo)(33.46 " sq units"