Question

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

Answer 1

side `a=sqrt(6)+sqrt(2)=3.863703305`
side `b=sqrt(6)+sqrt(2)=3.863703305`

Explanation:

The given parts of the triangle are
side `c=2`
Angle `A=(5pi)/12=75^@`
Angle `C=pi/6=30^@`

The third angle `B` can be readily solved

`A+B+C=180^@`

`75^@+B+30^@=180^@`

`B=180-105^@=75^@`

Therefore we have an isosceles triangle with angles A and B equal.

solve side `a` using Sine Law

`a/sin A=c/sin C`

`a=(c*sin A)/sin C=(2*sin 75^@)/(sin 30^@)=sqrt(6)+sqrt(2)`
by the definition of Isosceles triangle

`b=a=sqrt(6)+sqrt(2)`

We can check triangle using the Mollweide's Equation which involves all the 6 parts of the triangle

`(a-c)/b=sin (1/2(A-C))/(cos (1/2(B)))`

`(sqrt(6)+sqrt(2)-2)/(sqrt(6)+sqrt(2))=sin (1/2(75-30))/(cos (1/2(75)))`

`0.4823619098=0.4823619098`

This means all the 6 parts are correct.

God bless....I hope the explanation is useful.