application of fundamental theorem of integral calculus
We will derive the addition formulas of the sine and the cosine functions supposing known only their derivatives and the chain rule
.
But this expression is identically 0. By the fundamental theorem of integral calculus, must be a constant function. Since , we have
for any and naturally also for any . Because is a sum of two squares, the both addends of it have to vanish identically, which yields the equalities
These the addition formulas (http://planetmath.org/GoniometricFormulae)
Title | application of fundamental theorem of integral calculus |
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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 |