Sin A + Sin B + Sin C in a Triangle

Consider the triangle ABC and its circumcircle. Draw diameter of the circumcircle from B to A'.

Now, the angles at A and A' have the same measure.
Therefore sin A = sin A'.
  But  sin A = sin A' = a/2R
where R is the radius of the circumcircle.
Now, the same analysis works for sin B and sin C.   So,