Jordan's inequality

2 π x ≤ sin ⁡ ( x ) ≤ x  for  x ∈ [ 0 , π 2 ] {\displaystyle {\frac {2}{\pi }}x\leq \sin(x)\leq x{\text{ for }}x\in \left[0,{\frac {\pi }{2}}\right]}
unit circle with angle x and a second circle with radius | E G | = sin ⁡ ( x ) {\displaystyle |EG|=\sin(x)} around E. | D E | ≤ | D C ^ | ≤ | D G ^ | ⇔ sin ⁡ ( x ) ≤ x ≤ π 2 sin ⁡ ( x ) ⇒ 2 π x ≤ sin ⁡ ( x ) ≤ x {\displaystyle {\begin{aligned}&|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{aligned}}}

In mathematics, Jordan's inequality, named after Camille Jordan, states that[1]

2 π x ≤ sin ⁡ ( x ) ≤ x  for  x ∈ [ 0 , π 2 ] . {\displaystyle {\frac {2}{\pi }}x\leq \sin(x)\leq x{\text{ for }}x\in \left[0,{\frac {\pi }{2}}\right].}

It can be proven through the geometry of circles (see drawing).[2]

Notes

  1. ^ Weisstein, Eric W. "Jordan's inequality". MathWorld.
  2. ^ Feng Yuefeng, Proof without words: Jordan`s inequality, Mathematics Magazine, volume 69, no. 2, 1996, p. 126

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 Archived 2016-03-03 at the Wayback Machine. 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
  • Jordan's inequality at the Proof Wiki
  • Jordan's and Kober's inequalities at cut-the-knot.org
  • Weisstein, Eric W. "Jordan's inequality". MathWorld.