Jordan's Inequality

Theorem

 * JordansInequality.png


 * $\dfrac 2 \pi x \le \sin x \le x$

for all $x$ in the interval $\closedint 0 {\dfrac \pi 2}$

Proof
The inequality is true by Sine Inequality.

The inequality is true for $x = 0$ and $x = \dfrac \pi 2$, where we have equality.

Now consider $x \in \openint 0 {\dfrac \pi 2}$.

From Shape of Sine Function, $\sin x$ is concave on the interval $\closedint 0 \pi$.

Letting $x_1 = 0$, $x_2 = x$ and $x_3 = \dfrac \pi 2$ in the definition of Concave Real Function we obtain:


 * $\dfrac {\sin x} x \ge \dfrac {1 - \sin x} {\dfrac \pi 2 - x}$

Rearranging gives the desired inequality.