Talk:Trigonometric functions

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[edit] FA status

I think this is in need of review. One thing that strikes me immediately is the shortness of the lead section. I wouldn't even proceed with a GA assessment before the lead was expanded. Richard001 (talk) 09:07, 26 March 2008 (UTC)

[edit] Definitions

The article suggests several definitions of trigonometric funcitons, but does not provide any hint to equivalence of these definitions. Why should readers believe, that the function, defined with series, is periodic? dima (talk) 03:56, 25 May 2008 (UTC)

The normal approach in mathematical articles for such uncontroversial facts that are widely known among people familiar with mathematics, since they are taught in undergraduate courses and found in textbooks, is to include one or two references that cover the relevant standard material. I'm not sure the existing references in the article qualify. Alternatively, a proof might be included. I have not yet seen a good discussion of when to include proofs and when not; I'd hoped to work on that myself, but haven't yet found the time to do so. In this case the proof is not deep, but a good exposition may take up more space than is reasonable. If we can start from the fact that the derivative of the sine function is the cosine, and that of the cosine is minus the sine, it is easy to show that their Taylor series are as given.  --Lambiam 02:23, 26 May 2008 (UTC)
Yes, please do it. And periodicity too. dima (talk) 06:03, 26 May 2008 (UTC)
I've no plans to add a proof. As to periodicity: if the series definition and the geometric definition define the same function, then the function defined by the series definition is periodic because it is the same as the "geometric" function, which is obviously periodic.  --Lambiam 23:02, 26 May 2008 (UTC)
Lambiam, it is not obvious, because the "geometric" definition depends on the axioms of Euclidean geometry, so, the properties of trigonometric functions are posulated, not proven. dima (talk) 08:04, 27 May 2008 (UTC)
My statement was conditional (if ... then ...), and holds if the condition is satisfied. You are essentially saying that it is not obvious it is satisfied. It is well-known (and not difficult to prove) that the axioms of Euclid are satisfied in the plane formed by R2 endowed with the usual Euclidean metric, and to prove the addition formula you don't need to go beyond the constructible points, which are dense in R, so you don't even need to appeal to the Cantor-Dedekind axiom yet. If you now show that limx→0 (sin x)/x = 1 – something that goes beyond Euclid's axioms but should still hold in all models – you are basically done. As I said, a good exposition may take up more space than is reasonable.  --Lambiam 03:09, 28 May 2008 (UTC)
Lambiam, I was not clear, sorry. I wanted to say, that the "geometric definition" does not define any function until we accept the axiom about addition and measuring of angles. I could not understand that axiom, I felt myself stupid. Only after to deduce the periodicity of sin and cos from the differential equations, I realized teachers are wrong; they show things upsidedown. Correctly, first, there should be real numbers, then differential equations, then sin and cos, and only then the concept of angle has sense, it apears as atan2(y,x), and we can speak about constructible numbers; properties of elements in R2 can be deduced; there is no need to postulate them as axioms. The article should be rewritten; it should not postulate deducible things. dima (talk) 08:33, 28 May 2008 (UTC)
That goes beyond the purpose of an encyclopedia. We do not "postulate", we report established facts, which for mathematical statements should typically be either commonly accepted definitions or be deducible from such definitions. The relationship between Euclidean geometry and analytic geometry is well known and well established, and if one accepts the latter as the canonical model of the former (basically throwing in completion so as to be able to use analysis), the geometric and analytical definitions are provably equivalent, and it is a matter of convenience and taste which are taken as the definitions and which are deduced.  --Lambiam 17:46, 28 May 2008 (UTC)

[edit] Definitions using functional equations?

There is a section of the article claiming that sine and cosine are the unique functions satisfying the angle-addition formulas and a couple of other properties. I added a "citation needed" tag to this a while back, but no one has supplied a reference.

Besides the obvious need for a reference, I'm a little concerned that it may not be true. This definition seems closely related to the definition of the exponential function exp(ix) as the unique unit-magnitude function |f(x)|=1 that satisfies f(x+y)=f(x) f(y), plus some additional constraint to make sure you get exp(ix) and not exp(icx) for some c≠1. However, if you are not careful, such a definition of the exponential can be incomplete [Hewitt & Stromberg, Real and Abstract Analysis (Springer, 1965)]—you typically also need to require that the function be measurable, or alternatively require that it be anywhere-continuous, to exclude crazy non-measurable everywhere-discontinuous functions that can be defined satisfying that condition.

I'm wondering if some analogous requirement of measurability/continuity is required here, or if that is implicit in the inequality 0 < xcos(x) < sin(x) < x for 0 < x < 1.

—Steven G. Johnson (talk) 15:41, 12 June 2008 (UTC)

[edit] Periodic functions - problematic example

Is anyone else bothered by the square-wave example and the formula:

x_{\mathrm{square}}(t) = \frac{4}{\pi} \sum_{k=1}^\infty {\sin{\left ( (2k-1)t \right )}\over(2k-1)}

given in the article? This formula is rather subtle in its interpretation — it is true almost everywhere, but at the point of the discontinuity the series converges to zero regardless of how the square wave is defined at that point. Of course, if one redefines one's notion of a function to be a distribution, then strict equality holds, but otherwise it is a bit of an odd case.

Maybe it would be better to change this example to, e.g., a triangle wave, which converges pointwise everywhere and has no hidden caveats?

—Steven G. Johnson (talk) 15:49, 12 June 2008 (UTC)