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# Give a geometric proof that sin x < x for 0 < x < π/2

Consider the following figure: Compare the area of the triangle with the area of the sector to prove Further, prove, Proof. Assume the radius is . For we have the triangle has base length and height the area is The area of the circular sector is Since the area of the triangle is less than the area of the sector we have, Then, since we have for  for 1. tasos says: , but since the in the specified range we get , as a result, it doesn't justify . Of course, with its very easy to solve.

• RoRi says:

Hi, thanks! Yes, that was definitely incorrect. I put up a fix now, and also corrected the typo in the diagram that had the origin labelled with $P$.

• Anonymous says:

but we assume 0<x<pi/2 this would imply sinx<x for 0<x<pi which is not true

• Daniel says:

Your implication is false. x < pi/2 doesn't imply that pi/2 < x < pi, as you are implicitly stating by holding sin(x) < x for higher values of x.

It's like if I say "0 + x < 10 for 0 < x < 9" and you answer with "but this would imply that 0 + x < 10 for 0 < x < 548392 which is not true". Review your logic.