 Trigonometric inequality

An inequality involving trigonometric functions. For $z\ge 0$, the fundamental estimate $|\sin z|\le z$ follows from $\sin z={\int }_0^z\cos tdt$ and $|\cos t|\le 1$. Thus $1-\cos z=2{\sin }^2(z/2)\le z^2/2$, and integrating gives $0\le z-\sin z\le z^3/6$. For $0\le z\le \pi$, this implies $0\le z^2-{\sin }^{2}z\le z^4/3$. These bounds give explicit errors for central finite differences applied to sine modes.