Sine defined for a triangle inscribed in a circle with a diameter of one Let a circle be drawn with a diameter of one (and thus a radius of one half). Then let a triangle with vertices A, B, and C be inscribed in the circle (i.e. points A, B, and C are arbitrary points on the circle). 
Then a, the side of the triangle opposite angle A is equal to sin(A)
Likewise, b=sin(B) and c=sin(c). I have attempted to find or devise a proof of this, but I don't know where to start!

As lab bhattacharjee pointed out, what you just described is the law of sines, and it applies to all triangles. Google "law of sines proof" for more info.
