Question

A triangle has sides A, B, and C. Sides A and B have lengths of 2 and 5, respectively. The angle between A and C is `(11pi)/24` and the angle between B and C is `(7pi)/24`. What is the area of the triangle?

Answer 1

The triangle does not exist...
Here's the area anyways:
`A= (5sqrt2)/2 un^2 approx 3.54 un^2`

Explanation:

Area of the triangle:
`A= 1/2a*b*sinC`
Let's determine angle C:
`(24pi)/24-(7pi)/24-(11pi)/24= pi/4`

Area of the triangle:
`A= 1/2(2)(5)sin(pi/4)=`
`A= (5sqrt2)/2 un^2 approx 12.37 un^2`

How do I know this triangle particularly does not exist?
Well:
The law of sines states:
SinA/a= SinB/b
Therefore:
`B= arcsin((SinA*b)/a)`
`B= arcsin((Sin((7pi)/24)*5)/2)= undef.`
Since Angle B cannot be computed, this triangle does not exist