application of fundamental theorem of integral calculus

We will derive the addition formulasPlanetmathPlanetmath of the sine and the cosine functions supposing known only their derivatives and the chain ruleMathworldPlanetmath.

Define the functionMathworldPlanetmathF:  through


where α is, for the , a constant.  The derivative of F is easily calculated:
F(x)= 2[sinxcosα+cosxsinα-sin(x+α)][cosxcosα-sinxsinα-cos(x+α)]+2[cosxcosα-sinxsinα-cos(x+α)][-sinxcosα-cosxsinα+sin(x+α)]

But this expression is identically 0.  By the fundamental theorem of integral calculus, F must be a constant function.  Since  F(0)=0,  we have

F(x) 0

for any x and naturally also for any α.  Because F(x) is a sum of two squares, the both addends of it have to vanish identically, which yields the equalities

sinxcosα+cosxsinα-sin(x+α)= 0,cosxcosα-sinxsinα-cos(x+α)= 0.

These the addition formulas (

Title application of fundamental theorem of integral calculus
Canonical name ApplicationOfFundamentalTheoremOfIntegralCalculus
Date of creation 2013-03-22 18:50:52
Last modified on 2013-03-22 18:50:52
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Example
Classification msc 26A06
Related topic TrigonometricFormulasFromSeries