Question

A triangle has sides A, B, and C. The angle between sides A and B is `pi/8`. If side C has a length of `32` and the angle between sides B and C is `pi/12`, what is the length of side A?

Answer 1

`A=32sqrt(2-sqrt(3))/sqrt(2-sqrt(2))`

Explanation:

Angle measured `pi/12` lies across side `A`.

Angle measured `pi/8` lies across side `C=32`.

Using the Law of Sines, `A/sin(pi/12)=32/sin(pi/8)`

from which follows that `A=32sin(pi/12)/sin(pi/8)`

Let's determine the values of these two sines.
We will use the following trigonometric identities:
`cos(2x)=cos^2(x)-sin^2(x)=1-2sin^2(x)`
and, hence,
`sin^2(x)=(1-cos(2x))/2`

Using the above,

`sin(pi/12) = sqrt(sin^2(pi/12)) = sqrt((1-cos(pi/6))/2)=`
`=sqrt((2-sqrt(3))/4)=1/2sqrt(2-sqrt(3))`

`sin(pi/8) = sqrt(sin^2(pi/8)) = sqrt((1-cos(pi/4))/2) =`
`= sqrt((2-sqrt(2))/4)=1/2sqrt(2-sqrt(2))`

Therefore,
`A=32sqrt(2-sqrt(3))/sqrt(2-sqrt(2))`