Question

A triangle has sides A, B, and C. The angle between sides A and B is `pi/4`. If side C has a length of `5` and the angle between sides B and C is `pi/12`, what is the length of side A?

Answer 1

`A = C sina/sinc;`

substitute the values
`A = 5 sin15/sin60`
`and A = 5((2sqrt2 +sqrt3)/3)`
Plug in your calculator if you like
Hope it helps

Explanation:

Using the triangle angle theorem
we find the third angle is 120 thus

`180 = anglea+angleb+anglec`
Thus,
`anglea = 15; angleb=45 and anglec = 120`
Now you can use sin law to to compute side A
`A/sina = B/sinb =C/sinc`
Use the the 1st and third identities with respect to a, A, c, C
`A = C sina/sinc; substitute A = 5 sin15/sin120= 5 sin15/sin60=`
It uses the fact that `sin60 = sin120` and
and
`sin60 = (sqrt3)/2`
`sin15= ((sqrt3)+1)/((sqrt2)/2)` sothen