concavity of sine function

Theorem 1.

The sine function is concave on the interval [0,π].


Suppose that x and y lie in the interval [0,π/2]. Then sinx, siny, cosx, and cosy are all non-negative. Subtracting the identities




from each other, we conclude that


This implies that sin2x-sin2y0 if and only if cos2y-cos2x0, which is equivalentPlanetmathPlanetmath to stating that sin2xsin2y if and only if cos2xcos2y. Taking square roots, we conclude that sinxsiny if and only if cosxcosy.

Hence, we have


Multiply out both sides and move terms to conclude


Applying the angle addition and double-angle identities for the sine function, this becomes


This is equivalent to stating that, for all u,v[0,π],


which implies that sin is concave in the interval [0,π]. ∎

Title concavity of sine function
Canonical name ConcavityOfSineFunction
Date of creation 2013-03-22 17:00:26
Last modified on 2013-03-22 17:00:26
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Theorem
Classification msc 26A09
Classification msc 15-00