Trigonometric inequality
= Trigonometric inequality
An <inequality> involving <trigonometric functions>. For $z\geq0$, the fundamental estimate $|\sin z|\leq z$ follows from $\sin z=\int_0^z\cos t\,dt$ and $|\cos t|\leq1$. Thus $1-\cos z=2\sin^2(z/2)\leq z^2/2$, and integrating gives $0\leq z-\sin z\leq z^3/6$. For $0\leq z\leq\pi$, this implies $0\leq z^2-\sin^2z\leq z^4/3$. These bounds give explicit errors for <central finite differences> applied to <sine> modes.