More Jordan type inequalities

Abstract

The function $\tan (\pi x/2)/(\pi x/2)$ is expanded into a Laurent series of $1-x^2$, where the coefficients are given explicitly as combinations of zeta function of even integers. This is used to achieve a sequence of upper and lower bounds which are very precise even at the poles $x=1,-1$. Similar results are obtained for other trigonometric functions with poles.