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wu :: forums    riddles    putnam exam (pure math) (Moderators: Eigenray, Grimbal, william wu, towr, Icarus, SMQ)    Sinc(x) « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Sinc(x)  (Read 4664 times) ThudnBlunder Uberpuzzler The dewdrop slides into the shining Sea     Gender: Posts: 4489 Sinc(x)   « on: May 17th, 2008, 10:47am » Quote Modify Prove that sin(x)/x  =  cox(x/2)*cos(x/4)*cos(x/8)*cos(x/16) ................... IP Logged THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you. pex Uberpuzzler     Gender: Posts: 880 Re: Sinc(x)   « Reply #1 on: May 19th, 2008, 8:28am » Quote Modify 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). IP Logged ThudnBlunder Uberpuzzler The dewdrop slides into the shining Sea     Gender: Posts: 4489 Re: Sinc(x)   « Reply #2 on: May 19th, 2008, 10:52am » Quote Modify Yep, that's it, pex.   IP Logged THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you. william wu wu::riddles Administrator     Gender: Posts: 1291 Re: Sinc(x)   « Reply #3 on: May 20th, 2008, 1:44pm » Quote Modify There's a neat geometric interpretation of this formula in Eli Maor's book:   http://press.princeton.edu/books/maor/chapter_11.pdf « Last Edit: May 21st, 2008, 3:15pm by william wu » IP Logged [ wu ] : http://wuriddles.com / http://forums.wuriddles.com Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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