Question

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?

Answer 1

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-(11pi)/24-pi/8=(24-11-3)pi/24=(10pi)/24=(5pi)/12` and it has side `C` opposite it.

As side `A=3` has angle opposite it `pi/8` and `C` has opposite to it angle `(5pi)/12`. Now, using sine formula, we get

`3/sin(pi/8)=C/sin((5pi)/12)` or

`C=3xxsin((5pi)/12)/sin(pi/8)=3xx0.9659/0.3827=7.57`

Hence area of triangle is `1/2xx3xx7.57xxsin((11pi)/24)`

= `1/2xx3xx7.57xx0.9914=11.26`

We have not used `B=2` and as angle opposite it is `((11pi)/24)`, using sine formula

`2/sin((11pi)/24)=C/sin((5pi)/12)`

or `C=2xxsin((5pi)/12)/sin((11pi)/24)=2xx0.9659/0.9914=1.95`

and area of triangle is `1/2xx2xx1.95xxsin(pi/8)=1.95xx0.3827=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.