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"only", comments on linear diff eq proof of d(sin x)/dx = cos(x) |
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:I don't think your method at the other article is wrong...I think it's been misinterpreted. The point at which you use geometry and limits in one fell swoop is when you sneak in the result on the solutions of linear diff eqs. The problem here is that to prove (check) that this is the right solution requires knowing the derivatives of sine and cosine, so we're assuming what we're trying to prove. But, the argument is important and instructive. The diff eq itself along with the initial conditions can be "proven" informally using physics/vector ideas, (see Tristan Needham's book), i.e. the eqs come from a ''geometric'' conception of sine and cosine independent of analysis. Then, roughly the same argument (it's probably a bit different) will get you d(sin x)/dx = cos(x), using only properties of limits, or at worst, elementary properties of derivatives. Then, you have another "definition" of sine/cosine -- you define them as the solns of the IVP, and this definition is justified by the informal physics/vector analogy. It's an important way to look at it.
:[[User:Revolver|Revolver]] 09:26, 28 Jun 2004 (UTC)
Revolver: you are right, I misinterpreted. I thought you meant only infinite series and those two limits specifically on the trig identity page.--[[User:Dissipate|Dissipate]] 03:02, 29 Jun 2004 (UTC)
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