# A triangle has sides A, B, and C. Sides A and B have lengths of 12 and 15, respectively. The angle between A and C is (pi)/8 and the angle between B and C is  (7pi)/12. What is the area of the triangle?

Mar 26, 2018

color(brown)("Area of the triangle " A_t = (1/2) a * b * sin C = 71.4 " sq units"

#### Explanation:

$a = 12 , \hat{A} = \frac{7 \pi}{12} , b = 15 , \hat{B} = \frac{\pi}{8}$

To find the area of the triangle.

Third angle $\hat{C} = \pi - \frac{7 \pi}{12} - \frac{\pi}{8} = \frac{7 \pi}{24}$

Now we know, side a, side b and included ange C.

${A}_{t} = \left(\frac{1}{2}\right) a \cdot b \cdot \sin C = \left(\frac{1}{2}\right) \cdot 12 \cdot 15 \cdot \sin \left(\frac{7 \pi}{24}\right) = 71.4 \text{ sq units}$