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 December 12th, 2018, 01:26 PM #1 Member   Joined: Feb 2018 From: Canada Posts: 46 Thanks: 2 Richardson extrapolation Consider a regular n-sided polygon tangential to and enclosing the unit circle to approximate $\pi$. a) Use geometrical arguments to show that the half-perimeter of the polygon $C_n=n\tan(\pi /n) \geq \pi$. b) Calculate $C_4$ and $C_8$. c) Use Richardson extrapolation with $h = 1/n^2$ to improve upon your results. For part b) I've got: $C_4 = 4\tan(\frac{\pi}{4}) = 4\times 1 = 4,$ $C_8 = 8\tan(\frac{\pi}{8}) = 8 \frac{\sin(\frac{\pi}{8})}{\cos(\frac{\pi}{8})} = 8\frac{\frac{\sqrt{2-\sqrt{2}}}{2}}{\frac{\sqrt{2+\sqrt{2}}}{2}} = 8\frac{\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}$ By drawing, I have been able to do the part a) but I have no clue how to do part c). Can someone provide me some help? Thank you. Last edited by skipjack; December 12th, 2018 at 07:00 PM. December 12th, 2018, 01:29 PM #2 Senior Member   Joined: Aug 2012 Posts: 2,355 Thanks: 737 Today I learned. https://en.wikipedia.org/wiki/Richardson_extrapolation Thanks from topsquark and Shanonhaliwell December 12th, 2018, 07:04 PM #3 Global Moderator   Joined: Dec 2006 Posts: 20,921 Thanks: 2203 tan($\pi$/8) = √2 - 1 Tags extrapolation, richardson Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Dacu Algebra 3 July 30th, 2013 01:00 AM StevenDR Applied Math 1 April 9th, 2013 10:43 AM veronicak5678 Applied Math 0 October 31st, 2011 07:05 PM omaraansi Algebra 0 January 16th, 2011 11:44 AM cyberjaya Real Analysis 1 August 5th, 2010 03:53 PM

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