suppose the diameter of a circle has length d. the length of a chord is c and the length of the arc cut off by the cord is s. express sin(s/d) in terms of c and d
Let a be the angle subtending half the chord, and thus half the arc.
sin(a) = (c/2)/(d/2) = c/d
Now, s = r*a = d*a/2
so, s/d = a/2
sin(a/2) = sqrt(1-cos(a))/2
cos(a) = sqrt(1 - c^2/d^2)
sin(s/d) = sin(a/2) = sqrt(1 - sqrt(1 - c^2/d^2))/2
whew