Jordan's inequality

[math]\displaystyle{ \frac{2}{\pi}x\leq \sin(x) \leq x\text{ for }x \in \left[0,\frac{\pi}{2}\right] }[/math]

unit circle with angle x and a second circle with radius [math]\displaystyle{ |EG|=\sin(x) }[/math] around E. [math]\displaystyle{ \begin{align}&|DE|\leq|\widehat{DC}|\leq|\widehat{DG}|\\ \Leftrightarrow &\sin(x) \leq x \leq\tfrac{\pi}{2}\sin(x)\\ \Rightarrow &\tfrac{2}{\pi}x \leq \sin(x)\leq x \end{align} }[/math]

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

[math]\displaystyle{ \frac{2}{\pi}x\leq \sin(x) \leq x\text{ for }x \in \left[0,\frac{\pi}{2}\right]. }[/math]

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

Further reading

• Serge Colombo: Holomorphic Functions of One Variable. Taylor & Francis 1983, ISBN:0677059507, 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, ISSN 1223-7027
• Feng Qi: Jordan's Inequality: Refinements, Generealizations, Applications and related Problems . 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
• Weisstein, Eric W.. "Jordan's inequality". http://mathworld.wolfram.com/JordansInequality.html.