# A triangle has sides A, B, and C. The angle between sides A and B is pi/12 and the angle between sides B and C is pi/12. If side B has a length of 1, what is the area of the triangle?

Feb 17, 2018

Area of triangle Delta ABC = (1/2) a c sin B = color(blue)(0.134

#### Explanation:

Given $\hat{C} = \frac{\pi}{12} , \hat{A} = \frac{\pi}{12} , b = 1$

It’s an isosceles triangle with sides a & c equal, as their angles are equal.

$\hat{B} = \pi - \frac{\pi}{12} - \frac{\pi}{12} = \frac{5 \pi}{6}$

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

$\frac{a}{\sin} \left(\frac{\pi}{12}\right) = \frac{c}{\sin} \left(\frac{\pi}{12}\right) = \frac{1}{\sin} \left(\frac{5 \pi}{6}\right)$

$a = c = \frac{1 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{5 \pi}{6}\right) = 0.5176$

Area of triangle $\Delta A B C = \left(\frac{1}{2}\right) a c \sin B$

=> (1/2) * 0.5176 * 0.5176 * sin ((5pi)/6) = color(blue)(0.134