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# Prove the inequality 2x/π < sin x < x for 0 < x < π/2

Prove the inequality Proof. Define a function . Then, since for we have Since and is increasing on (since its derivative is positive) we have For the inequality on the left (which is much more subtle), we want to show So, we consider the function We know and Now, if we can show that is decreasing on the whole interval then we will be done (since this would mean on the whole interval since if it were less than somewhere then it would have to increase to get back to on the right end of the interval).

To show is decreasing on the whole interval we will show that its derivative is negative. To that end, define a function (This is the numerator in the expression we got for . Since we know the denominator of that expression is always positive, we are going to show this is always negative to conclude is always negative.) Now, and for . Thus, is negative, and so is decreasing. Since we then have is negative on the whole interval. Therefore, on the interval. Hence, is decreasing. Hence, we indeed have for all ### One comment

1. Brahim says:

Very nice proof. Many thanks