Talk:List of trigonometric identities

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[edit] arctan(y, x)

where arctan(y, x) is the generalization of arctan(y/x) which covers the entire circular range (see also
the account of this same identity in "symmetry, periodicity, and shifts" above for this generalization of arctan).

I believe this may be referencing section 3.3 "Shifts"; however the correlation does not appear to be explicit -- please expound either section as necessary and clarify the notation arctan(b,a) which does not seem to appear anywhere else in the article.

--Eibwen 08:23, 13 April 2006 (UTC)

Just a few weeks ago when I found this page, this section was very clear about what to use for the phase (something like: if a > 0 it's one thing, and if a <= 0 it's another). Now I'm not sure how to interpret it, because there now seems to be a discrepancy between what is stated in the section Other sums of trigonometric functions and what is stated in Periodicity, symmetry, and shifts. I lack the confidence to determine which is correct, so I'm not going to touch it, but I would genuinely like to see this resolved. --Qrystal 12:44, 15 November 2006 (UTC)

[edit] What about the Harmonic Addition Theorem and the Prosthaphaeresis Formulas

The list is incomplete!

--Eibwen 03:03, 26 March 2006 (UTC)


Those identities are in this article and have been there for a long time. If you can't find them here, you haven't looked very hard at all. Michael Hardy 23:59, 26 March 2006 (UTC)


Upon further review I've found both -- however I'd assert that the organization of the article is not readily amenable to finding a particular formula. To elaborate:

I learned the "Sum-to-product identities" as the "trigonometric sum" formulas, consequently I wrongly presumed their absence because I did not anticipate the inclusion of the "ouput" of the identity ("-to-product") in the heading. Similarly, I did not expect to find the harmonic sum identity under the vague heading "Other sums of trigonometric functions". While I did skim the article itself looking for the formulas, it was not until I thoroughly read the article (knowing this time that they were present) that I found the identities I was looking for.

Some basic reorganization could make this article much more accessible. To illustrate:


  • Trigonemetic Sum Identities
    • Angle Sum and Difference Identities
      • Geometric proofs
    • Sum to product identities
    • Harmonic Addition Identity
    • Multiple-angle formulae
    • Inverse trigonometric functions
  • Trigonemetic Product Identities
    • Product-to-sum identities
    • Half-angle formulae
    • Double-angle formulae
    • Triple-angle formulae
    • Infinite product formulae
  • Exponential Trigonometric Identities
    • Pythagorean identities
    • Power-reduction formulae
    • Exponential forms


While not perfect (and incomplete) I'd find such organization much more amenable to finding a particular identity.

--Eibwen 08:23, 13 April 2006 (UTC)

The Pythagorean identities seem more basic than others that preceed them in your list. And why call them "exponential"? An exponential function is a function is which the input variable is the exponent. I have no idea what identities you're calling "exponential forms".
Also, you should not have so many capital letters in section headings; see Wikipedia:Manual of Style. "Angle sum and difference identities" conforms to Wikipedia's conventions; "Angle Sum and Difference Identities" does not. Michael Hardy 21:14, 13 April 2006 (UTC)

[edit] Identity vs Equation

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)
I hope you will all excuse my intrusion (I have yet to regularly log in before posting.) It strikes me that there are two trivial symmetry relations missing from this page, namely sin(x + Pi) = -sin(x) and cos(x + Pi) = -cos(x), of course, the plus signs should be plus-minus signs. --Dwee

[edit] 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)

[edit] 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)

Sverdrup: I disagree. If we don't add any more proofs then I think it should stay the way it is. But if we add any more then I agree, proofs should have their own article.--Dissipate 19:15, 29 Jun 2004 (UTC)

Laura Anderson: Just one month ago, this article was shorter and more easily navigated. It's way too long now. We should definitely add links to other articles and focus on the identities themselves (sum to product, etc.)

[edit] Relating trig functions

Regarding:

One procedure that can be used to obtain the elements of this table is as follows:...Third, solve this equation for φ(arcψ(x)) and that is the answer to the question.

There's nothing wrong with this, but the way I usually teach it is not algebraically but geometrically. Draw a right triangle (imagined to be inscribed in the first quadrant of unit circle). Label the unknown angle θ or something. Say you want to find cos(arcsin(x)). Then θ is the angle which gives you a sine of x, so put an x on the vertical leg and a 1 on the hypotenuse. The cosine of this angle θ is then the horizontal leg over the hypotenuse. Use the pythagorean theorem to get the horizontal leg equal to sqrt(1-x^2). This has the advantage that the only identity to remember is the pythagorean theorem (the others are just different versions, anyhow) and drawing a picture seems easier to visualise what's going on. Of course, to explain this would require pictures, which I don't know how to create or upload. Revolver 23:33, 25 Sep 2004 (UTC)

[edit] sin-1

The notation sin-1 is not rigorous because the function sin is not invertible. So if there is no objection, i will erase it . Mr Spok.

sin-1 is generally used in most modern text books to mean arcsine, many students are taught to use sin-1 this page could be confusing for them without this.

sin-1 is standard notation, and Wikipedia should follow it. Charles Matthews 21:33, 12 Apr 2005 (UTC)
Except that it should be written as sin−1 rather than sin-1, i.e. with a proper minus sign rather than a stubby little hyphen (or in TeX as \sin^{-1}\,\!. Especially in subscripts and superscripts, a hyphen can be hard to see. Michael Hardy 21:51, 12 Apr 2005 (UTC)
sin−1 certainly is standard notation, but it does create confusion with 1/sin. I consider safer to use arcsin to avoid any misunderstanding. --Doctor C 20:00, 4 August 2006 (UTC)

So if the sentence remains the same and sin-1 is changed to sin−1 that would satisfy everyone?

[edit] sin−1

sin−1 does not equal 1/sin−1

They are fundamentaly different sin−1 is the inverse function not the recipricol.

This is fundamental!

Nor is sin−1 equal to 1/sin. Michael Hardy 20:45, 4 August 2006 (UTC)

[edit] trigonometric identity

I entirely disagree with your introduction of radian measure instead of degrees (or evern in addition to degrees) in the "identities without variables" section. Before getting into my reasons (which I would have thought were obvious) may I ask why, if you thought radians should be added, you did not change the surrounding text in the appropriate way? The text ceased to make sense because of your changes. Michael Hardy 20:36, 21 July 2005 (UTC)

Michael, Well I've looked and the only mention of degrees vs radians in that paragraph is:
Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators
The expressions that you deleted were in radians and were correct. I disagree that there is any benefit (felicity??) in having degrees to the exclusion of radians - they are equivalent for most purposes (with radians being necessary for some). Since much of the article is couched in radians or implied radians, my additions were for consistency. Having gone to the trouble of making my improvements once, I will not revert your deletions, even though I think the article is the poorer for them. Ian Cairns 22:32, 21 July 2005 (UTC)

In the first place, just in case anyone reading this leaps to conclusions, I was not opposing the use of radians in general, and obviously radian measure is indispensible in such things as (d/dx) sin(x) = cos(x). But when I read

\cos 20^\circ\cdot\cos 40^\circ\cdot\cos 80^\circ=\frac{1}{8}

I can feel the pattern effortlessly in a way that doesn't happen when I see radians. I suspect that is because the numbers are integers. If that fails to happen for you, that at least surprises me. Michael Hardy 23:34, 29 July 2005 (UTC)

[edit] all functiion of logarithim

[edit] Pythagorean identities

Are they definitely correct? I thought 1/(cosx)^2 was secx, not cscx. The same goes for 1/(sinx)^2 being cscx, not secx.

1/cos2(x) is sec2(x), not just sec(x). Similarly, 1/sin2(x) = csc2(s), not csc(x). The formulas look correct to me. How did you conclude that anyone was mistaking 1/cos for csc? Michael Hardy 21:34, 9 June 2006 (UTC)

[edit] Why introduce cis in the "Angle sum and differences identities" section?

  • First off, why define cis when it simply refers to e^ix?
  • Also, why introduce a complex function in an article that pretty much only deals with trig functions in the real case?
  • And even more so, why introduce cis when nothing is ever done with it? Although it is said that it can prove the aforementioned identites, there is no proof provided.

Basically, the entry has no reason for introducing the cis function within an article about real trig identities so I think it should be either removed or it's more familiar form, e^ix, should be used to prove a trig identity (or maybe even just provide a link to a section on euler's formula and include the proof for a trig identity there).

I was puzzled by this "cis" thing, myself, having never heard of it referred to in this way. It's a good thing this was mentioned in talk, because I'm going to take the initiative and remove all mention of "cis". The section they were in (Angle sum and difference identities) already has a link to Euler's formula, as recommended in the comment above; however, there is also a link to Ptolemy's theorem, which I don't see as being relevant at all (except that the identities are used in it) so I am removing the "see also".
I do disagree with the statement that complex functions need not be present in this article. However, I believe the link between trigonometric and exponential forms is sufficient in the section entitled Exponential forms, though I have added a link there to Euler's formula as well. I support the recommendation that proofs of trig identities using Euler's formula should be included, perhaps at Proofs of trigonometric identities.
--Qrystal 13:40, 15 November 2006 (UTC)

It's been a somewhat standard thing for many decades, and there's a reason for using it in some contexts rather than eix. The reason is that in certain expository and pedogical contexts one wishes to work with that function without saying in advance that it's an exponential function, and work one's way through the reasoning whose ultimate conclusion is that it's an exponential function. Michael Hardy 22:26, 15 November 2006 (UTC)

[edit] New section: why "cis"?

I've just added a new section answering the question raised in the header above. Michael Hardy 00:23, 22 November 2006 (UTC)

I guess I missed it, as it's been removed already. I reviewed the history, and found it there... and I also found it interesting, so I tried to see if it was relocated to somewhere else. The cis article has mention of the use of cis in mathematics, and it refers to Euler's formula, which doesn't mention cis at all. It would be a shame to see the information from this "new (but removed) section" be lost; perhaps this historical/pedagogical topic warrants its own page? --Qrystal 15:38, 8 December 2006 (UTC)

Perhaps. I've just reinserted it. An anonymous editor who never did any other Wikiepedia edits had removed it. Michael Hardy 20:17, 8 December 2006 (UTC)

[edit] Identity symbol

If these are identities, shouldn't the dominant symbol in the math formulae be the identity symbol, rather than the equals sign? 82.12.107.150 16:28, 1 October 2006 (UTC)

  • Are you referring to the three bar equal sign? If so, that's the symbol for definition (or modular equivalence, but not identity) and these aren't definitions, just things that can be proven true from the definitions. If not, I don't remember any identity sign, so could you elaborate? MagiMaster 17:45, 1 October 2006 (UTC)

[edit] Merge the proofs of the identities to Proofs of trigonometric identities

There's no need to clutter up this already haphazard article with proofs. --Ķĩřβȳ♥ŤįɱéØ 10:50, 20 October 2006 (UTC)

[edit] Trigonometric conversions table errors

sin(arctan(x)) = x/(1+x^2)

The table shows x/(1-x^2) I havn't checked for further errors

Reference wolfram's mathworld under InverseTrigonometricFunctions

I think I've cleaned up the table. Please note: Entries in any column, not in any row, are the six functions of one angle. Michael Hardy 23:39, 21 November 2006 (UTC)
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