Question

A triangle has sides A, B, and C. If the angle between sides A and B is `(pi)/6`, the angle between sides B and C is `(5pi)/12`, and the length of B is 2, what is the area of the triangle?

Answer 1

`Area=1.93184` square units

Explanation:

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" `/_ A` and angle between side "c" and "a" by `/_ B`.

Note:- the sign `/_` is read as "angle".
We are given with `/_C` and `/_A`. We can calculate `/_B` by using the fact that the sum of any triangles' interior angels is pi radian.
`implies /_A+/_B+/_C=pi`
`implies pi/6+/_B+(5pi)/12=pi`
`implies/_B=pi-(7pi)/12=(5pi)/12`
`implies /_B=(5pi)/12`

It is given that side `b=2.`

Using Law of Sines
`(Sin/_B)/b=(sin/_C)/c`
`implies (Sin((5pi)/12))/2=sin((5pi)/12)/c`
`implies 1/2=1/c`
`implies c=2`

Therefore, side `c=2`

Area is also given by
`Area=1/2bcSin/_A=1/2*2*2Sin((7pi)/12)=2*0.96592=1.93184`square units
`implies Area=1.93184` square units