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This statement is false. I show a proof for this that does not use geometry or properties of limits on the trig identity article. I am removing the word only. --[[User:Dissipate|Dissipate]] 06:11, 28 Jun 2004 (UTC)
:I assume you're talking about the "linear differential equations" approach to prove d(sin x)/dx = cos(x). I have some comments about that, but first, I would point out that I think you misread what I wrote. If you read it carefully, all that it claims is that "There exists a method which shows that the derivative of sine is cosine and of cosine is negative sine, and which only uses geometry and the properties of limits". I made no such claim that this method was ITSELF the only method to solve the problem. I only made an existence statement, not a uniqueness statement.
:But it doesn't matter much, because even the proof you suggest uses geometry and limits. Moreover, ANY PROOF MUST USE EACH OF THESE, for the simple reasons (1) if sine and cosine are to be defined indepedently of infinite series, or analytic methods, say, then they have to be defined geometrically; in my method, they are the real and imaginary parts of a point on the unit circle (or, x- and y-coordinates) parametrised by the circle's arc length, (2) the problem asks us to find a derivative...since a derivative is defined using limits, by definition we must use limits at some point.
: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)
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