Question

A triangle has sides A, B, and C. The angle between sides A and B is `pi/3`. If side C has a length of `12` and the angle between sides B and C is `pi/12`, what is the length of side A?

Answer 1

`2 sqrt(6) (sqrt(3)-1)`

Explanation:

Assuming angles opposite to sides `A, B` and `C` are `/_A, /_B and /_C`, respectively.

Then
`/_C = pi/3 and /_A = pi/12`

Using Sine Rule

`(Sin/_A)/A = (Sin/_B)/B = (Sin/_C)/C`

we have,
`(Sin/_A)/A = (Sin/_C)/C`
`(Sin(pi/12))/A = (Sin(pi/3))/12`
`A=(sqrt(3)-1)/(2 sqrt(2))*12*1/(sqrt3/2)`
`or, A=2 sqrt(6) (sqrt(3)-1)`
`or, A~~3.586`