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Sine functions and intervals

Source: 1st Brazilian MO, 1979, Problem 1 (from Kalva backup)

December 23, 2017
Brazilian Math OlympiadtrigonometryfunctioninequalitiesalgebraBrazilian Math Olympiad 1979

Problem Statement

Show that if a<ba < b are in the interval [0,π2]\left[0, \frac{\pi}{2}\right] then a−sin⁡a<b−sin⁡ba - \sin a < b - \sin b. Is this true for a<ba < b in the interval [π,3π2]\left[\pi,\frac{3\pi}{2}\right]?
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