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Friday, July 17, 2015

Indian Statistical Institute (ISI) B.Math & B.Stat : Trigonometry

Indian Statistical Institute B.Math & B.Stat Solved Problems, Vinod Singh ~ Kolkata Find the ratio of the areas of the regular pentagons inscribed and circumscribed around a given circle. Let a be the side of the circumscribed pentagon and b be that of the inscribed pentagon. First note that for the circle is inscribed for the exterior pentagon and circumscribed for the interior pentagon. Therefore the inradius of the exterior polygon, say r is equal to the circumradius, say R of the interior pentagon, i.e., R=r. See the figure below. Using standard formula, a=2rtanπ5,b=2Rsinπ5. Area of a regular polygon having n sides is n×(side)24cotπn. Therefore the required ratio is (5×b24cotπ55×a24cotπ5)=b2a2=(2Rsinπ5)2(2rtanπ5)2=cos2π5

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