Jordan's Inequality

Author

Eric W. Weisstein
April 14, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/JordansInequality.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/JordansInequality.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Jordan's Inequality

Plot[{2x/Pi,Sin[x],x},{x,0,Pi/2},PlotStyle->{Blue,Black,Red},
AxesOrigin->{0,0},
Ticks->{Range[0,4]Pi/8,Automatic},
AxesLabel->TraditionalForm/@{x,f[x]}]

Proof

Resolve[ForAll[x,0<x<Pi/2,2x/Pi<x]]

Resolve[ForAll[x,0<x<Pi/2,2x/Pi<Sin[x]<x]]

FullSimplify[2x/Pi≤Sin[x]<=x,0<=x<=Pi/2]