Jordan's inequality

$\frac{2}{\pi }x\le \sin (x)\le x\text{ for }x\in [0,\frac{\pi }{2}]$

unit circle with angle x and a second circle with radius $|EG|=\sin (x)$ around E. $\begin{array}{cc}& |DE|\le |\widehat{DC}|\le |\widehat{DG}|\\ \iff & \sin (x)\le x\le \frac{\pi }{2}\sin (x)\\ \Rightarrow & \frac{2}{\pi }x\le \sin (x)\le x\end{array}$

In mathematics, Jordan's inequality, named after Camille Jordan, states that

$\frac{2}{\pi }x\le \sin (x)\le x\text{ for }x\in [0,\frac{\pi }{2}].$

It can be proven through the geometry of circles (see drawing).

Further reading

• Serge Colombo: Holomorphic Functions of One Variable. Taylor & Francis 1983, Template:ISBN, p. 167-168 (online copy)
• Da-Wei Niu, Jian Cao, Feng Qi: Generealizations of Jordan's Inequality and Concerned Relations. U.P.B. Sci. Bull., Series A, Volume 72, Issue 3, 2010, Template:Issn
• Feng Qi: Jordan's Inequality: Refinements, Generealizations, Applications and related Problems Template:Webarchive. RGMIA Res Rep Coll (2006), Volume: 9, Issue: 3, Pages: 243–259
• Meng-Kuang Kuo: Refinements of Jordan's inequality. Journal of Inequalities and Applications 2011, 2011:130, doi:10.1186/1029-242X-2011-130

External links

• Jordan's inequality at the Proof Wiki
• Jordan's and Kober's inequalities at cut-the-knot.org
• Template:MathWorld