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Sides of a triangle

Source:

May 12, 2006
probabilitytrigonometrysymmetryinequalitiesalgebrafunctiondomain

Problem Statement

An angle xx is chosen at random from the interval 0∘<x<90∘0^\circ < x < 90^\circ. Let pp be the probability that the numbers sin⁡2x\sin^2 x, cos⁡2x\cos^2 x, and sin⁡xcos⁡x\sin x \cos x are not the lengths of the sides of a triangle. Given that p=d/np = d/n, where dd is the number of degrees in arctan⁡m\arctan m and mm and nn are positive integers with m+n<1000m + n < 1000, find m+nm + n.
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