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# General trig identities of form f(ab)

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I''m trying to derive the area formula of a circle from a general n-gon area formula: AC = lim(n->+inf)[ nr2sin(pi/n)cos(pi/n) So, the problem is, how can one split up the parameter of a trig function (spec. sin and cos) such that it is possible to simplify it by solving the limit? This is not homework, just something for personal satisfaction. I''ve tried searching Google, but didn''t find anything after a cursory search for trig identities relating to products. Thanks in advance, ZE. //email me.//zealouselixir software.//msdn.//n00biez.//

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quote:
Original post by ZealousElixir
I''m trying to derive the area formula of a circle from a general n-gon area formula:

AC = lim(n->+inf)[ nr2sin(pi/n)cos(pi/n)

To do this I would apply the limit first rather than trying to mess around with trig identities...

= r^2 lim(n->+inf) [ n sin(pi/n) cos(pi/n) ]
let t = 1/n:
= r^2 lim(n->+inf) [ n ] lim(t->0)[sin(pi*t) cos(pi*t)]
= r^2 lim(n->+inf) (n) lim(t->0) [pi * t * 1]
= r^2 * pi * (n*t*1)
= pi r^2, OMG!

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quote:
Original post by ZealousElixir
This is not homework, just something for personal satisfaction.

That''s fine, . But, please in the future show us your own work to solve the problem rather than just telling us you searched other sources for the answer first. That would provide some comfort that you''re not just saying its not homework to pacify our rage!

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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I''ll scan in my final analysis tonight. Sorry for not providing better evidence beforehand that illustrated my progress. This question spawned out of a discussion with a friend in Cal class, but was totally unrelated to the subject matter (indefinite integrals), and since school is pretty boring, I made it my task of the day to derive the formula of a circle from the formula for a general regular n-gon.

Thanks for the replies,
ZE.

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