derivatives of sine and cosine

The of the derivatives of sine and cosine is a bit simpler by using the prosthaphaeresis formulasMathworldPlanetmath

sinα-sinβ= 2sin(α-β2)cos(α+β2), (1)
cosα-cosβ=-2sin(α+β2)sin(α-β2). (2)

Let x,t be any real numbers such that  tx.  Then we obtain

sinx-sintx-t=2sin(x-t2)cos(x+t2)x-t=sin(x-t2)(x-t2)cos(x+t2) 1cos(x+x2)=cosx,

as  tx.  Here we used the known limit  limu0sinuu=1  (see this entry (

The derivative of cosine is calculated similarly:


as  tx.

Title derivatives of sine and cosine
Canonical name DerivativesOfSineAndCosine
Date of creation 2013-03-22 16:58:58
Last modified on 2013-03-22 16:58:58
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 10
Author Wkbj79 (1863)
Entry type Derivation
Classification msc 26A09
Related topic DerivativesOfSinXAndCosX
Related topic LimitOfDisplaystyleFracsinXxAsXApproaches0
Related topic DefinitionsInTrigonometry
Related topic LimitRulesOfFunctions