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| sin m \theta_2 - sin m \theta_1| <= m| sin (\theta_2 - \theta_1)|

Source: Flanders Math Olympiad 2012 p3

December 24, 2022
trigonometryinequalitiesalgebra

Problem Statement

(a) Show that for any angle θ\theta and for any natural number mm: ∣sin⁡mθ∣≤m∣sin⁡θ∣| \sin m\theta| \le m| \sin \theta|
(b) Show that for all angles θ1\theta_1 and θ2\theta_2 and for all even natural numbers mm: ∣sin⁡mθ2−sin⁡mθ1∣≤m∣sin⁡(θ2−θ1)∣| \sin m \theta_2 - \sin m \theta_1| \le m| \sin (\theta_2 - \theta_1)|
(c) Show that for every odd natural number mm there are two angles, resp. θ1\theta_1 and θ2\theta_2, exist for which the inequality in (b) is not valid.
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