A triangle has sides A,B, and C. If the angle between sides A and B is (7pi)/8, the angle between sides B and C is pi/12, and the length of B is 12, what is the area of the triangle?

Apr 7, 2018

color(green)("Area of " Delta " " A_t = 54.62 " sq units"

Explanation:

$\hat{A} = \frac{\pi}{12} , \hat{C} = \frac{7 \pi}{8} , b = 12 , \text{ To find the area of } \Delta$

$\hat{B} = \pi - \hat{A} - \hat{C} = \pi - \frac{\pi}{12} - \frac{7 \pi}{8} = \frac{\pi}{24}$

Applying the Law of Sines,

$\frac{a}{\sin} \left(\frac{\pi}{12}\right) = \frac{12}{\sin} \left(\frac{\pi}{24}\right) = \frac{c}{\sin} \left(\frac{7 \pi}{8}\right)$

$a = \frac{12 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{\pi}{24}\right) = 23.79$

Knowing two sides a,b and the included angle C, to find the area we can use the formula color(crimson)(A_t = (1/2) a b sin C

A_t = (1/2) * 23.79 * 12 * sin((7pi)/8) = color(purple)(54.62 " sq units"