Lim(x*sqrt(1-cos(2*Pi/x)),x->Infinty) using analytical methods

[dipstik]

Hey guys,

My cousin came over and was talking about his geometry class and it got me to derive 2*Pi*r and Pi*r^2 using polygons of n sides as n goes to infinity. For the area I ended up using a taylor expansion of sin to weasel myself out of a 0 times infinity, but this is not being very nice. I have tried three iteration of L'Hopital and ended up with a tangent, with more trig functions to come from further iterations I think. The limit converges nicely, pretty quickly too, but I don't want to use numerical methods. I tried a binomial expansion but ended up with a long list of 0*inf. terms there.

Any ideas?

 

[MrJB]

You have Cos(2*stuff) so use a double angle trig identity to get something squared in the sqrt.