Question

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

Answer 1

First of all let me denote the sides with small letters `a`, `b` and `c`.
Let me name the angle between side `a` and `b` by `/_ C`, angle between side `b` and `c` by `/_ A` and angle between side `c` and `a` by `/_ B`.

Note:- the sign `/_` is read as "angle".
We are given with `/_B` and `/_A`. We can calculate `/_C` by using the fact that the sum of any triangles' interior angels is `pi` radian.
`implies /_A+/_B+/_C=pi`
`implies (11pi)/24+(11pi)/24+/_C=pi`
`implies/_C=pi-((11pi)/24+(11pi)/24)=pi-(11pi)/12=pi/12`
`implies /_C=pi/12`

It is given that side `a=7` and side `b=2.`

Area is also given by
`Area=1/2a*bSin/_C`

`implies Area=1/2*7*2Sin(pi/12)=7*0.2588=1.8116` square units
`implies Area=1.8116` square units