# A triangle has sides A, B, and C. Sides A and B have lengths of 3 and 2, respectively. The angle between A and C is (11pi)/24 and the angle between B and C is  (pi)/8. What is the area of the triangle?

May 20, 2016

If $A = 3$ then area is $7.57$ and if $B = 2$ area is $0.746$

#### Explanation:

The third angle opposite sides $A$ and $B$ is

$\pi - \frac{11 \pi}{24} - \frac{\pi}{8} = \left(24 - 11 - 3\right) \frac{\pi}{24} = \frac{10 \pi}{24} = \frac{5 \pi}{12}$ and it has side $C$ opposite it.

As side $A = 3$ has angle opposite it $\frac{\pi}{8}$ and $C$ has opposite to it angle $\frac{5 \pi}{12}$. Now, using sine formula, we get

$\frac{3}{\sin} \left(\frac{\pi}{8}\right) = \frac{C}{\sin} \left(\frac{5 \pi}{12}\right)$ or

$C = 3 \times \sin \frac{\frac{5 \pi}{12}}{\sin} \left(\frac{\pi}{8}\right) = 3 \times \frac{0.9659}{0.3827} = 7.57$

Hence area of triangle is $\frac{1}{2} \times 3 \times 7.57 \times \sin \left(\frac{11 \pi}{24}\right)$

= $\frac{1}{2} \times 3 \times 7.57 \times 0.9914 = 11.26$

We have not used $B = 2$ and as angle opposite it is $\left(\frac{11 \pi}{24}\right)$, using sine formula

$\frac{2}{\sin} \left(\frac{11 \pi}{24}\right) = \frac{C}{\sin} \left(\frac{5 \pi}{12}\right)$

or $C = 2 \times \sin \frac{\frac{5 \pi}{12}}{\sin} \left(\frac{11 \pi}{24}\right) = 2 \times \frac{0.9659}{0.9914} = 1.95$

and area of triangle is $\frac{1}{2} \times 2 \times 1.95 \times \sin \left(\frac{\pi}{8}\right) = 1.95 \times 0.3827 = 0.746$

Why this dichotomy? The fact is that we need either (a) one side and both angles on it; or (b) two sides and included angle and (iii) three sides of a triangle to identify a triangle and find area or other sides and angles of a triangle. However here we have been given four parameters and they give two different results depending on whether we take side $A = 3$ or $B = 2$ into consideration. In short, given three angles (third is derivable from other two), the two sides are not compatible and in fact refer to two different triangles.