# A triangle has sides A, B, and C. The angle between sides A and B is (pi)/2 and the angle between sides B and C is pi/12. If side B has a length of 22, what is the area of the triangle?

Mar 26, 2018

Area of the rt. triangle color(maroon)(A_t = (1/2) * a * b = 64.9 " sq units"

#### Explanation:

$b = 22 , \hat{A} = \frac{\pi}{12} , \hat{C} = \frac{\pi}{2}$

$\hat{B} = \pi - \hat{A} - \hat{\mathbb{C}} = \pi - \frac{\pi}{12} - \frac{\pi}{2} = \frac{5 \pi}{12}$

Applying Law of sines,

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

$a = \frac{b \sin A}{\sin} B = \frac{22 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{5 \pi}{12}\right) = 5.9$

It's a right triangle with $\hat{C} = \frac{\pi}{2}$

Hence area of the rt. triangle ${A}_{t} = \left(\frac{1}{2}\right) \cdot a \cdot b = \left(\frac{1}{2}\right) \cdot 5.9 \cdot 22 = 64.9 \text{ sq units}$