Jordan's Inequality


Jordan's inequality states that for

Prove Jordan's Inequality.


 What does this inequality say?     We have a  curve,   y = sin x, placed between two linear functions in the range .      The inequality is reversed in the range .     One interpretation of this is that in this range, the value of sin x is always less than the curve tangent to the sine curce at the origin (y = x)   and it always lies above the chord of the sine curve between the origin and   

So what?   Why is the Jordan Inequality important in  pure or applied mathematics?

Who was Jordan?



HINTS for thinking about a proof:

Consider a unit circle, OA = 1 with a point P on the circle. Construct a perpendicular from P to the x-axis with M being the foot of the perpendicular and Q its reflection in the x-axis. Let x be the measure of the angle POM. The line PM has length sin x.



Construct a circle of radius MP and center at M.


Find elements in the figure that reflect what you are to prove in Jordan's Inequality.




Reference: Yuefeng, Feng. (1996) Proof without words. Mathematics Magazine. 69, p. 126.

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