# A triangle has sides A,B, and C. If the angle between sides A and B is (pi)/4, the angle between sides B and C is pi/6, and the length of B is 1, what is the area of the triangle?

Aug 10, 2018

color(indigo)(A_t = 1/2 a b sin C ~~ 0.183, " sq. units"

#### Explanation:

$\hat{A} = \frac{\pi}{6} , \hat{C} = \frac{\pi}{4} , \hat{B} = \frac{7 \pi}{12} , b = 1$

Law of Sines : $\frac{a}{\sin} A = \frac{b}{\sin} b = \frac{c}{\sin} C$

$a = \frac{b \cdot \sin A}{\sin} B = \frac{1 \cdot \sin \left(\frac{\pi}{6}\right)}{\sin} \left(\frac{7 \pi}{12}\right)$

$a \approx 0.5176$

$\text{Area of D} \Delta = {A}_{t} = \frac{1}{2} a b \sin C$

color(indigo)(A_t = 1/2 * 0.5176 * 1 * sin (pi/4) ~~ 0.183, " sq. units"