Talk:List of trigonometric identities

Trigononmetric identities should not be called "equations". "Equation" means something like "3x + 5 = 17", which is to be solved for x. "Identity" means something like (x + 1)² = x² + 2x + 1, which is true of all values of x, and is not to be solved for x, but rather is to be proved, if one is to contemplate something to be done with it. "Equality" is a more general term that includes both equations and identities. Michael Hardy 19:18 27 May 2003 (UTC)

I am the one who changed the first sentence slightly because I thought we usually identities as a set so to me identities are bahaha sounds more natural. The current version seems fine to me. -- Taku 20:18 27 May 2003 (UTC)

Linear DEs proof of d(sin x)/dx = cos x

The proof is still not correct, for the same reasons I said at Talk:Trigonometric function. I will come back later and try to repair it. One obvious new mistake -- f + g = h + j, does not imply f = h or f = j, this doesn't work even with numbers, 1 + 1 = 0 + 2. Here's a sketch of the way it could go, if you want to try this:

Let E(t) = C(t) + iS(t) be the parametric equation describing the motion of a particle on the unit circle in the complex plane (and yes, I do mean E(t), NOT E(it)). Here, t is a real variable, E is a complex function, and C and S are real functions (the real and imaginary parts of E). We assume that the motion of the particle is parametrised by the arc length of the circle. This conforms to our geometric notion of cosine and sine as the x- and y-coordinates of a point on the unit circle. (The only difference is, here, it's being traced out over time t). From this, we can deduce that the particle is moving at unit speed. To see why this is so, note that the ratio of the distance between 2 points on the circle, one measured along a straight line, the other along the circular arc connecting them, tends to 1 as the points tend to each other. (Draw a picture with a right triangle of very small angle, basically this amounts to sin x is roughtly = x for small x.) This is more or less where the "limit" argument comes in. What about the direction of velocity vector? It's not hard to see this must be always at a 90 degree angle to the position vector, just by the geometry of the situation (here's where geometry comes in). So, the velocity vector always has equal length (1) to position vector, and is rotated 90 degrees CCW. These two changes correspond to multiplying by i. So, dE/dt = iE, or dC/dt + i(dS/dt) = iE = −S + iC. Rquating real and imaginary parts, dS/dt = C and dC/dt = −S. The second-order equation with initial values for sine and cosine is encapsulated in the single equation d2E/dE2 + E = 0, with E(0) = 1 and E'(0) = i. Physically, this is just an expression of centripetal force, and you can define sine and cosine to be the imaginary and real parts of the unique solution of this equation. This second-order equation can be solved just as the first-order above, because it can be decoupled into 2 first-order equations which are essentially identical to solve.

Revolver 21:14, 28 Jun 2004 (UTC)

Revolver: you are right. When my DE teacher showed us these proofs he didn't show the part where you actually equate the S(x) and C(x) functions with the actual solutions. So I tried to do it makeshift, if you could fix it that would be cool. In the meantime I'm going to try one other way. If its wrong, just correct it. --Dissipate 02:56, 29 Jun 2004 (UTC)

No problem...that's what the editing process is for. I did read the new change, it's still not quite right; here, the problem is that the set of solutions is all linear combinations of sines and cosines, not just multiples of sine or cosine. (Geometrically, you've taken a 2-dim soln space and reduced it to the union of 2 lines). I'll come up with something in the near future along the lines of what I have above -- the physics way of seeing it is what's important, I'll try to mention that. I'll try to keep what you already have, but don't take it personally if I rearrange things quite a bit. Revolver 05:12, 29 Jun 2004 (UTC)

I also think it would be good to really stress how all these identities are much easier to understand and prove with aid of complex numbers. Mention of Euler's formula and DeMoivre's formula is given, but a more comprehensive treatment of this approach would be useful. Revolver 05:19, 29 Jun 2004 (UTC)

This article need some cleanup

We should really just list the identities, and link to the relevant proofs, if they are not very brief. Sections like the new proof that dsin(x)=dxcos(x) and the geometric proof really have no place here. [[User:Sverdrup|✏ SverdrupSverdrup]] 08:27, 29 Jun 2004 (UTC)

Revolver: I just found out that I was making the DE proofs much more complicated than necessary. Read this short .PDF on the matter and tell me what you think. Rigorous Definitions of Sine and Cosine. Apparently it is much better & easier to show that sine and cosine are solutions right off the bat instead of coming up with abstract functions and proving properties of those. In my opinion I think we should change to this format for the DE proofs.--Dissipate 10:33, 29 Jun 2004 (UTC)

Ok, I just changed the DE proof section to what I mentioned. I think I helped us avoid a lot of confusion/complexity in addition to making the proof correct. --Dissipate 11:41, 29 Jun 2004 (UTC)

I read the .pdf file (well,...skimmed). It's pretty clear and correct, there are only a couple questions I would have. First, the approach taken there is essentially what I wrote in a section on the Trigonometric function article, so perhaps these can be coordinated somehow. As to the paper, there was really only one point I don't agree with. With all due respect to Buck (a respected author), I don't think using the "arc-length parameter" definition for cosine and sine is illogical or circular reasoning. It's true, the arc-length parameter definition depends on having a well-defined notion of arc-length or length of a rectifiable curve, but it's not necessary to measure lengths on this curve. Think of it like this, we can prove that the integral

${\mbox{Arc length}}=L(x)=\int _{s=x}^{s=1}(1-s^{2})^{-1/2}ds$

is a strictly decreasing function of x on [0, 1], we don't know what L(0) is, except that it's > 0, and we know L(1) = 0. The intermediate value theorem says that for any value t between L(0) and 0, there's an x in (0, 1) with L(x) = t. Now, define cosine(t) = x and sine(t) = sqrt(1 − x^2). Then cosine, sine are well-defined, parametrised by arc length, and conform to our prior geometric "definition" of them. And we never had to actually evaluate an arc length integral at all!

The approach of proving identities from the 2nd order DE is instructive, it's clear and elegant. I didn't really care for the Sturm-Liouville stuff the author did just to get periodicity and the definition of &pi. (Which, essentially, amounts to "proving" that our functions really are what we think they should be.) It's much easier to define &pi as say, twice the first positive root of the cosine function. And periodicity is much easier to see by passing to the complex exponential from the beginning. I mean, by encapsulating the two 2nd order equations into a single complex DE. This not only cuts repetition, it gives the geometric interpretation in terms of velocity and acceleration.

In general, it's much better and more correct. The only things I might do differently (personal preference) would be to present the 2nd order DE in terms of complex functions (I realise this may not help people who don't know complex numbers), and then get the relation to geometry by defining &pi in a nicer way. (A lot of this approach is in the prologue to Rudin, Real and Complex Analysis). Of course, I think it's important to realise that NONE of these approaches is "the best approach" and when trying to explain or understand something, getting as many different explanations as possible is a good thing. It's starting to intrude on the article, though, maybe move these to "Proofs involving trigonometric functions and identities" or something similar.

Revolver 13:05, 29 Jun 2004 (UTC)

Revolver: I didn't even read that part of the article. Do you think it is important to include material from that section as well? It seems like pretty heady material.--Dissipate 19:08, 29 Jun 2004 (UTC)