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Topic: Sinc(x) (Read 4668 times) |
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ThudnBlunder
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Prove that sin(x)/x = cox(x/2)*cos(x/4)*cos(x/8)*cos(x/16) ...................
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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pex
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Re: Sinc(x)
« Reply #1 on: May 19th, 2008, 8:28am » |
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on May 17th, 2008, 10:47am, ThudanBlunder wrote:Prove that sin(x)/x = cox(x/2)*cos(x/4)*cos(x/8)*cos(x/16) ................... |
| hidden: | 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) = 2n sin(x/2n) * product[k=1..n] cos(x/2k) Thus, for all positive integers n, sin(x) / x = sin(x/2n) / (x/2n) * product[k=1..n] cos(x/2k) Taking limits, we have lim[n->inf] sin(x/2n) / (x/2n) = lim[t->0] sin(t) / t = 1 and therefore sin(x) / x = product[k=1..inf] cos(x/2k). |
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ThudnBlunder
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Re: Sinc(x)
« Reply #2 on: May 19th, 2008, 10:52am » |
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Yep, that's it, pex.
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