On natural approaches related to classical trigonometric inequalities
Abstract:In this paper, we establish sharp inequalities for trigonometric functions. We prove in particular for 0<x<π2 and any n≥5 0<Pn(x) < (sinx)2−x3cotx<Pn−1(x)+[(2π)2n−n−1∑k=3ak(2π)2n−2k]x2n where Pn(x)=∑n3=kakx2k+1 is a n-polynomial, with positive coefficients (k≥5), ak=22k−2 (2k−2)!(|B2k−2|+(−1)k+1(2k−1)k), B2k are Bernoulli numbers. This improves a lot of lower bounds of sin(x)x and generalizes inequalities chains. Moreover, bounds are obtained for other trigonometric inequalities as Huygens and Cusa inequalities as well as for the function gn(x)=(sin(x)x)2(1−2(2xπ)2n+21−(2xπ)2)+tan(x)x, n≥1.