# A triangle has sides A, B, and C. If the angle between sides A and B is (pi)/8, the angle between sides B and C is (5pi)/6, and the length of side B is 19, what is the area of the triangle?

Feb 17, 2018

Area of triangle A_t = (1/2) a b sin C ~~ ~color(brown)( 264.5996 sq units

#### Explanation:

Given $\hat{C} = \frac{\pi}{8} , \hat{A} = \frac{5 \pi}{6} , b = 19$

Third angle $\hat{B} = \pi - \left(\frac{5 \pi}{6} + \frac{\pi}{8}\right) = \frac{\pi}{24}$

$\frac{a}{\sin} A = \frac{b}{\sin} B = \frac{c}{\sin} C$

a = (19 * sin ((5pi)/6)) / sin (pi/24)~~color(blue)( 72.7823

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

=> (1/2) * 19 * 72.7823 * sin((pi/8) ~~color(brown)( 264.5996