A triangle has sides A,B, and C. If the angle between sides A and B is (2pi)/3, the angle between sides B and C is pi/6, and the length of B is 2, what is the area of the triangle?

Feb 17, 2018

A_b = color(blue)(1.732

Explanation:

Sum of the three angles of a triangle $= {\pi}^{c}$

Hence the third angle $\hat{B} = \pi - \frac{2 \pi}{3} - \frac{\pi}{6} = \frac{\pi}{6}$

It’s an isosceles triangle with sides a & b equal.

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

When length 2 corresponds to $\angle B = \angle \left(\frac{\pi}{6}\right)$

$\frac{a}{\sin} \left(\frac{\pi}{6}\right) = \frac{2}{\sin} \left(\frac{\pi}{6}\right) = \frac{c}{\sin} \left(\frac{2 \pi}{3}\right)$

$\therefore a = 2 , c = \frac{2 \sin \left(\frac{2 \pi}{3}\right)}{\sin} \left(\frac{\pi}{6}\right) = 3.4641$

Area of triangle $A - b = \left(\frac{1}{2}\right) \cdot a c \sin B = \left(\frac{1}{2}\right) \cdot 2 \cdot 3.4641 \cdot \sin \left(\frac{\pi}{6}\right) = 1.732$