

Title: Sinc(x) Post by ThudanBlunder on May 17^{th}, 2008, 10:47am Prove that sin(x)/x = cox(x/2)*cos(x/4)*cos(x/8)*cos(x/16) ................... 

Title: Re: Sinc(x) Post by pex on May 19^{th}, 2008, 8:28am on 05/17/08 at 10:47:32, ThudanBlunder wrote:
[hideb] Repeatedly applying sin(2t) = 2 cos(t) sin(t), we find sin(x) = 2 cos(x/2) sin(x/2) sin(x) = 4 cos(x/2) cos(x/4) sin(x/4) sin(x) = 8 cos(x/2) cos(x/4) cos(x/8) sin(x/8) ... sin(x) = 2^{n} sin(x/2^{n}) * product[k=1..n] cos(x/2^{k}) Thus, for all positive integers n, sin(x) / x = sin(x/2^{n}) / (x/2^{n}) * product[k=1..n] cos(x/2^{k}) Taking limits, we have lim[n>inf] sin(x/2^{n}) / (x/2^{n}) = lim[t>0] sin(t) / t = 1 and therefore sin(x) / x = product[k=1..inf] cos(x/2^{k}). [/hideb] 

Title: Re: Sinc(x) Post by ThudanBlunder on May 19^{th}, 2008, 10:52am Yep, that's it, pex. :) 

Title: Re: Sinc(x) Post by william wu on May 20^{th}, 2008, 1:44pm There's a neat geometric interpretation of this formula in Eli Maor's book: http://press.princeton.edu/books/maor/chapter_11.pdf 

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