Very nice! Even if there may exist a more intuitive proof, this is very short and very clear, and it avoids actually doing any nasty definite integration. —Dan ----- On 19/09/2020 22:19, Dan Asimov wrote:
Wolfram alpha tells me that to 100 digits, the geometric mean of the values of sin(x) for 0 ≤ x ≤ π is exactly 1/2.
Can someone prove this? Even better, is there an intuitive reason that it ought to be true?
The following things are clearly equal; call them all M. - GM of sin x on (0,π). - GM of sin x on (0,π/2). - GM of cos x on (0,π/2). Second times third equals M^2 but also equals GM of 1/2 sin 2x on (0,π/2) equals GM of 1/2 sin x on (0,π) equals M/2. So M^2 = M/2 and so (modulo checking that M isn't zero) M = 1/2. This is a very short proof but I'm not sure it counts as an intuitive reason. Maybe there's some geometrical way to think about sin 2x = 2 sin x cos x that would bridge the gap? -----