Suppose we have the N vertices of a regular N-gon inscribed in a circle of radius 1. Select one vertex W and draw line segments from W to each of the other N-1 vertices. What is the total product of the lengths of these line segments? (old problem - original author unknown)

By the product formula given, the formula would products of sines from zero to pi/2 (or 180 degrees). I have the idea that an inductive technique could be used, as sin(pi/2 - x) = cos (x). Then two sines can be combined into one, as sin (2x) = 2 sin(x) cos(x), or sin(2x) = 2 sin(x)sin(pi/2 - x), thereby reducing the number of intervals or the value of N, to half the original.  This would work for values of N that are powers of 2 (using induction multiplicatively rather than additively), but I can't see how to formalize this, or where then N comes from in the formula. (I see where the halving of the intervals offsets the powers of two.)

Posted by Charlie on 2005-10-20 14:21:02