Talk:List of trigonometric identities/Archive 1

Archive 1

some of these are horribly hard to read.

Would:

cos(x/2) = √ [ ( 1 + cos(x) ) / 2 ]

be easier?

What about dropping the simple brackets, eg cosx instead of cos(x)

Different types or sizes of brackets would definitely help. I don't like dropping the brackets around arguments, because then people start writing things like cos 2x or cos (x+y)/2 and it is completely unclear what is meant. AxelBoldt

Unsorted text

This page was MUCH better before the useless introduction of organizing tables - now it's hard to find the equation one needs and it's missing information.

Tagging

What's with the tagging? I don't think such tags make sense for basic science articles, while i agree that even for common knowledge some reference math books could/should be listed, but it often makes absolutely no sense to work with quotes/inline citation or footnotes.--Kmhkmh 16:32, 22 September 2007 (UTC)

Missing/Deleted identities ?

Where did the Sum-to-product identities(such as sin(x)+sin(y)) go? Was there any reason to delete them? It looks like the page has undergone some massive editing this year and imho it has become worse. Imho the primary purpose of the article should be to give a correct and somewhat comprehensive list of all trigometric identities out there, which are of interest to various groups (mathematicians,engineers, laymen,students). This also means identities should not be removed because they are "just" a special case of a more general case or the can "easily" derived from another. Please keep in mind the different purposes and groups using this page as a lookup/reference. I don't mind adding additional infos/improving the format but that should not result in identities getting deleted. --Kmhkmh 16:47, 24 May 2007 (UTC)

I agree completely. And this cis stuff does not belong here, one line identifying cis(x) = cos x + isin(x) is plenty for this page. Linking to a seperate article is better. --24.207.160.213 20:58, 19 August 2007 (UTC)

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)

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)

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

dear dwee, I agree with you. sin(x + Pi) seems missing. But from the text above the symmetry section it looks like we were only going to look at where reflections ..resulted in another trig function.. ala cos( Pi/2 - theta) = sin theta. That's all that is justified from the sentence above the table. I wish that sentence said ..here are some fun reflections.. then we could have our sin(x + Pi). 68.13.126.138 (talk) 05:53, 1 February 2010 (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. ✏ 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.)

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)

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?

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)

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)

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)

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 e^ix. 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)

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)

I agree with the anonymous user: the whole cis() dissertation seems to me to have little to do with trig identities. I suppose you could make cis its own entry and list its raison d'etre there, but I think the identities page should stay more focused on identities. Derekt75 12:05, 15 June 2007 (UTC)

Also agree: the discussion might be interesting or useful, but does not belong on this page. A link to its relocated address would suffice. By the by: I happen to think that cis(•) is a useful notation. —DIV (128.250.204.118 07:31, 13 August 2007 (UTC))

cis has its own page. If someone wants to add the history or the pedagogy there, I won't complain. To me, cis(x) is a definition, not a trig identity, so I don't think it should be here. sinc(x) is also a useful notation, but is (rightfully) not listed. cis(x) is actually still listed under exponential definitions. So, after over a year of "discussion" on this, I've been bold and removed the sections. Derekt75 (talk) 22:54, 18 March 2008 (UTC)

It would seem someone has recovered the deleted cis entry in the exponential definitions. It is also mentioned in the "Extension of half-angle formulae" section just below. I think both references should be removed, as there is no other mention of cis on this page. And4e (talk) 22:14, 10 November 2009 (UTC)

I see cis mentioned there, but that's all. No one's put back the discussion of it, which explained why one would want to temporarily conceal the fact that it's an exponential function for pedagogical reasons. Michael Hardy (talk) 04:44, 11 November 2009 (UTC)

cis is rather obscure and little known, though it has a certain interest. I don't agree at all with the opinions raised above that one should not admit identities involving complex numbers: why not? I don't see that "trigonometric identity" has to mean "real trigonometric identity". When I checked the article a few minutes ago there was just one passing mention of "cis", with no explanation, which seemed pointless. I have added a few words explaining what it is an abbreviation for, but I have not added any more information about it. I have gone back into the history of the article and read the deleted section on cis. I find it interesting. It was deleted with the edit summary I completely removed cis. This is the consensus I see in the discussion page from over a year ago. However, having read the above discussion I do not see consensus to delete it: I see differing opinions. I wonder about reinstating it. JamesBWatson (talk) 09:46, 12 November 2009 (UTC)

I agree with Derekt75 above; I think the the information about cis is interesting, but should be put into a separate cis_(mathematics) page. It is pretty off topic for a page about trig identities. If that page existed, then I would feel better about cis being mentioned in this article (with a link to the cis article). Otherwise I think the introduction of the cis notation here isn't justified. And4e (talk) 02:41, 19 November 2009 (UTC)

Cis is not much used, and I could see some sense in suggesting it be excluded for that reason. However, what little use it has is entirely to do with trigonometric identities, so I don't see why it is off topic. It is used (or at least was within living memory, as I remember it being used when I was at school), whereas I believe that hacovercosine (for example) is long since obsolete, but as far as I know nobody has suggested removing all mention of that. The idea of having a separate article on cis seems to me to be giving it far more prominence than it deserves, and a fairly brief mention here for anyone who wants to know what it means seems to make much more sense. JamesBWatson (talk) 20:55, 19 November 2009 (UTC)

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)

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)

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)

You could eliminate the confusion by placing phi to the left of the table, and psi on top of it. That, combined with the phi(arcpsi(x)) should make it clear whether or not you look in the row or column first. Would the table still work if the functions on the top row all had arc- added to the beginning of them? —The preceding unsigned comment was added by 134.253.26.11 (talk) 22:41, 2 May 2007 (UTC).

Redundant parentheses

A common practice in Mathematics is to write sin x instead of sin (x) and sin 2x instead of sin (2x). Shouldn't the formulas in this article be written without the unnecessary parentheses? — Telking 09:57, 27 February 2007 (UTC)

That's a common practice, but in many cases I, for one, prefer to write these with parentheses. Michael Hardy 20:08, 27 February 2007 (UTC)

I prefer leaving out parens myself, but other peple (such as Mr Hardy, here) may find the other easier to write and read. OneWeirdDude 23:18, 5 March 2007 (UTC)

Considering the common use of powers and the very formal, reference-like tone of the article, parentheses should be used, especially because these are functions. I rarely see f(x) written as fx is a formal setting. —The preceding unsigned comment was added by 134.253.26.11 (talk) 22:28, 2 May 2007 (UTC).

Redirect

Shouldn't "trig identity" and "trigonometric identities" redirect here? Ketsuekigata 14:59, 23 April 2007 (UTC)

It appears that those two pages have redirected to this page since May 2006. Michael Hardy 15:46, 23 April 2007 (UTC)

Ah, how odd. It didn't do that the first time for me. Ketsuekigata 19:42, 23 April 2007 (UTC)

circular function picture

Ive always thought that the tangent line began at the point 1,0 as opposed to where it is now. Can someone explain it to me? —The preceding unsigned comment was added by 124.190.128.160 (talk) 12:50, 27 April 2007 (UTC).

There are different ways of doing it. I prefer the more conventional way to the one in this picture. Maybe I'll put together a different one at some point. Michael Hardy 18:09, 4 May 2007 (UTC)

Page needs help!

This page is a sprawling mess of different forms of equations and identities and I think that the best way to help this article is to simplify and shorten it. I feel that, as a list, it should not contain long proofs;and that, as a trig article it should not contain any hyperbolic functions. Maybe we should create a parallel article List of hyperbolic identities. Any help would be appreciated. Conrad.Irwin 23:08, 30 April 2007 (UTC)

Hard to find specific identities

Instead of providing proofs, why not just list the identities organized by type (half-angle, power-reduction, etc.). All I really want to see when searching for "trig identities" is a bunch of identities that aren't conjested with proofs. Links to proofs would make the page more compact and a better reference piece. Perhaps separate pages titled "Table of trigonometric function identities" and "Proofs and derivations of..." is required. —The preceding unsigned comment was added by 134.253.26.11 (talk) 22:25, 2 May 2007 (UTC).

I fully agree and even worse some identities seem to be missing now (so you won't find them at all) --Kmhkmh 16:51, 24 May 2007 (UTC)

square identities

I can find the formula $\sin ^{2}x+\cos ^{2}x=1$ in the page but i can't find the formula $\cot ^{2}x+1=\csc ^{2}x$. —The preceding unsigned comment was added by Novwik (talk • contribs) 11:01, 6 May 2007 (UTC).

Derivations

Some of the derivations on the page seem to use $\theta$ as a "number", and use x as an angle. This means you end up taking the arc-whatever of an angle, to get an number. The trigonometric functions take an angle as their argument and return a "number" (for want of a better word), and the inverse trig functions take a "number" as their argument and return an angle. It seems odd, when everywhere else, $\theta$ is the angle, and x is the "number". Why is $\theta$ not being used in its conventional meaning? Am I missing something here? If nobody objects, I'll change this. Alternatively, if I forget to change it, which is quite likely, feel free to change it. Tim Goodwyn 17:17, 19 May 2007 (UTC)

Are you referring only to the identities in the section titled "exponential definitions"? Michael Hardy 20:10, 19 May 2007 (UTC)

reorganize

This article is not as good as it once was.

Why are the two identities in "Basic relationships" grouped together into a single section containing nothing else? It doesn't make sense. Michael Hardy 22:25, 24 May 2007 (UTC)

Definitely needs to be split/reorganized. Is the a way we could make a variety of article subpages or sublists of derivations so that the load time for the page is less than 30-45seconds. Perhaps a brief explanation of the identities then a link to the relevant sub page or list.--Cronholm144 11:43, 25 May 2007 (UTC)

Probably derivations should be on a separate page. Also, I'd consider excluding identities that are about hyperbolic functions only. Michael Hardy 20:40, 25 May 2007 (UTC)

Removed Content

I have removed a lot of the content from this article because it didnt fit. What I have removed can be found at Talk:List of trigonometric identities/removed with reasons for why it doesnt fit. I can forsee there being a lot of disagreement about this, but I felt that without some drastic action, this article would take too much effort to tidy up. Conrad.Irwin 22:16, 7 June 2007 (UTC)

Whew - way to be bold! This looks like a well-executed, healthy trim of the content. Nice work. Doctormatt 22:46, 7 June 2007 (UTC)

I think it was a good idea to make the move. Having a separate apge for proofs or other things is good idea. Though it might be justified to add 1 or 2 things again, i'd overall it looks clearly better now.--Kmhkmh 14:55, 8 June 2007 (UTC)

printability

This article is very printer unfriendly. Many of the equations get cut off, and the PNG images are printed as solid black squares. &#9992; James C. (talk) 20:23, 9 February 2008 (UTC)

Power-reduction formulas

I suggest to add general power-reduction formulas.
I'm not sure, but I think they are something like this:
$cos(x)^{2k-1}=\Sigma _{j=1}^{k}\left[{{\left(_{k-j}^{2k-1}\right)} \over {2^{2k-2}}}\cdot cos((2j-1)x)\right]$
Where $\left(_{k-j}^{2k-1}\right)$ is binomial coefficient.
There are also formulas for $sin(x)^{2k-1},\;cos(x)^{2k},\;sin(x)^{2k}$ and perhaps more. They all can be derivered from $(e^{ix}\pm e^{-ix})^{n}$ —Preceding unsigned comment added by 79.183.155.81 (talk • contribs) 20:15, 3 June 2008

Sum of sines and cosines with arguments in arithmetic progression

There is no proof of these identities on wikipedia or anywhere else on the web. Can anyone verify/prove them? —Preceding unsigned comment added by 69.136.78.92 (talk) 21:21, 15 November 2008 (UTC)

Not registered != vandalism

I'm rather annoyed that Pharaoh of the Wizards is reverting my edits, here and elsewhere.

Currently under "Matrix form" there's a main article link to matrix multiplication. This implies that matrix multiplication is a page filling in the details about the matrix form of the sum and difference formula. Rot. It's related, nothing more. Call off your dogs. 150.203.35.113 (talk) 08:26, 18 November 2008 (UTC)

Actually, we could link to rotation matrices, whose multiplication results in a rotation of the sum of the angles of the two rotations. 190.231.116.145 (talk) 23:41, 20 July 2009 (UTC)

Table formatting

[Moved from User talk:Ozob. Ozob (talk) 22:44, 20 December 2008 (UTC)]

[1] I disagree. Provide reasoning as to why it's clearer. It looks to me as though there are more functions than there are. The parenthesis makes it clear what are the abbreviations. —Anonymous Dissident^Talk 05:12, 20 December 2008 (UTC)

Just for everyone else, here's the version of the table that I prefer:

Function		Inverse function		Reciprocal		Inverse reciprocal
sine	sin	arcsine	arcsin	cosecant	csc	arccosecant	arccsc
cosine	cos	arccosine	arccos	secant	sec	arcsecant	arcsec
tangent	tan	arctangent	arctan	cotangent	cot	arccotangent	arccot

and as Anonymous Dissident prefers:

Function	Inverse function	Reciprocal	Inverse reciprocal
sine (sin)	arcsine (arcsin)	cosecant (csc)	arccosecant (arccsc)
cosine (cos)	arccosine (arccos)	secant (sec)	arcsecant (arcsec)
tangent (tan)	arctangent (arctan)	cotangent (cot)	arccotangent (arccot)

My feeling is that parentheses don't accurately indicate what the relationship between the function name and the abbreviation. I could imagine someone who had never encountered these functions before being puzzled as to what the function is called: Is it "sine" or "sin"? Or if the person is confused about function notation, like so many students are, they might even think it should be sine(sin). Separating the two names into two table entries avoids this. I also think it makes the individual table entries easier to read.

However, when I first saw your change, my impulse was not to revert but to try to think of a way to indicate on the table that one column is the full name and the next is the abbreviation. I think that would be a better solution if we could do it. But I couldn't come up with anything better than adding a lot of new columns called "Full name" and "Abbreviation", and that seemed like it would come out worse than either of the above two. To be honest I don't know how to make this table look good and be clear. If you have any ideas, I'd love to hear them. Ozob (talk) 22:44, 20 December 2008 (UTC)

Parentheses are often used to indicate an abbreviation:

The Federal Bureau of Investigation (FBI) is an agency that blah blah blah blah....

If someone seeing the parentheses could be confused about the relationship between the function and its abbreviation, then someone seeing the more elaborate table could be just as confused or more so. Parentheses imply a sort of subordinate role rather than an equal footing, and that helps make the meaning clear. Michael Hardy (talk) 22:57, 20 December 2008 (UTC)

Perhaps the best thing, then, is to change the text prior to the table to either

To avoid the ambiguity of the notation ⁻¹, which is used for both reciprocals and inverses, the reciprocals and inverses of trigonometric functions are given distinctive names. In the table below, each pair of entries gives the full name of the function followed by its most common abbreviation. The longer form 'cosec' is sometimes used for the cosecant function in place of 'csc'.

or

To avoid the ambiguity of the notation ⁻¹, which is used for both reciprocals and inverses, the reciprocals and inverses of trigonometric functions are given distinctive names. In the table below, the full name of the function is given first and its most common abbreviation is given in parentheses. The longer form 'cosec' is sometimes used for the cosecant function in place of 'csc'.

depending on which table we use. Or it might be better to put the description of how to read the table in a caption. As long as there's a description like that I'm okay with Anonymous Dissident's proposed table. Ozob (talk) 03:17, 21 December 2008 (UTC)

More Inverse Formulas

I feel there should be more formulas for the inverse functions. I'm certain there should be some type of formula for arcsin(2x) and arcsin(x/2), for instance, assuming you know arcsin(x) (and arcsine(2x) for the second). If there isn't, someone should try to find one. And if it's transcedental for rational X, then I feel that math will have lost some of it's beauty. Timeroot (talk) 04:34, 8 June 2009 (UTC)

If x = 1/2, then arcsin(2x) is π/2. Ozob (talk) 20:01, 8 June 2009 (UTC)

Well, that one's pretty obvious. What I mean is, if arcsin(x)=y, and you know x and y, what is arcsin(2x) in general (in terms of x and y?) Also, what's arcsin(x/2)? arcsin(3x)? arcsin(x/3)? For instnace, since we know the arcsin(1/2)=pi/6, and arcsin(2/2)=arcsin(1)=pi/6, what's arcsin((1/2)*(1/2))=arcsin(1/4)? Timeroot (talk)

If you want arcsin(2x) in terms of x and y, rather than just in terms of y, then that's really easy. Here it is: arcsin(2x). Michael Hardy (talk) 00:17, 21 July 2009 (UTC)

I'm sorry if I don't find that humor hilarious... I was honestly wondering. In terms of x and y... without using logarithims, any arc-triginometric function, or any area-hyperbolic function. That should cover it... I mean something like "2*sqrt(x)*y or whatever. (Just as an example) Timeroot (talk) 05:42, 26 August 2009 (UTC)

I don't think it was meant to be humourous: it was a perfectly serious answer. It seems that Timeroot meant something different from what he/she said. The new version of the question is, unfortunately, not clearly enough specified, because it leaves open so large a range of functions that it is trivially easy to define an ad hoc function to do the job. This is not some kind of pedantry or joke: to be able to give a useful meaning to the question it is necessary to define precisely what functions are permitted. Probably the set of algebraic functions pretty well covers what Timeroot has in mind. If so the answer is pretty certainly "no". JamesBWatson (talk) 08:56, 27 August 2009 (UTC)

Shortly after I wrote the above I had another thought, and tried to add a note about it. Unfortunately there was then a problem with the Wikipedia servers and I lost connection. I made a note of what I had tried to post, meaning to try again the next day. I have only just found my note again. Despite the time that has gone, and the likelihood that the person who raised the matter has long since lost interest, I thought I might as well post it anyway. Someone may find it of interest.

I can't have been thinking very hard when I wrote that comment, or I wouldn't have written "pretty certainly". The question "arcsin(x)=y, and you know x and y, what is arcsin(2x)" is equivalent to "solve sin $z$ = 2 sin $y$ for $z$". Since that equation has in general infinitely many solutions, and an algebraic equation can have only finitely many, no algebraic function can possibly do the job. JamesBWatson (talk) 14:31, 20 January 2010 (UTC)

Hi. I had not lost interest in this, as a matter of fact. Thank you for bringing it to my attention. I'm sorry if I misinterpreted the response, or if I wasn't clear enough. I suppose one could say I'm looking for the "simplest" way to define these functions, where addition adds a very small level of complexity, and where something like a radical adds more complexity. And I didn't intend to have the original function back - I was hoping for something longer perhaps, but something that can be done with less advanced functions. I accept that there may be no solution, but I would like to point out that if sin(y)=x, then also sin(y+2*k*pi)=x and sin(pi-y+2*k*pi)=x. Using these formulas, given one solution, all can be found. So theoretically, an algebraic could exist.

Another possibility that may provide some useful insight is phrasing it in terms of a simple limit, preferably without any intermediate complex number. For example, the natural logarithim of x is the limit (as N->Infinity) of N*( x^(1/N) - x^(-1/N) )/2. If something relatively simple like that existed to describe the function arcsin(2*x) or arcsin(2*sin(x)), that would certainly seem good. And I appreciate the interest with this! :-) Timeroot (talk) 08:49, 31 January 2010 (UTC)

"So theoretically, an algebraic could exist": no. The functions involved are all analytic functions, and, informally speaking, if two analytic functions are equal in a domain then they are essentially the same function everywhere, in the sense that they can both be analytically extended to be the same. The solution can be given in terms of arcsin by a relation (or, if you prefer "multi-valued function") which is analytic. Let's call that function/relation f. If we found an algebraic relation (say g) giving particular solutions, i.e. a relation which, over a domain, gives values equal to particular values of f, then g could by analytic continuation be extended to agree with f completely, and therefore to give infinitely many solutions. However, an analytic continuation of an algebraic function is still algebraic, and therefore can have only finitely many values, giving a contradiction. Without the restriction to analytic functions/relations there would be no such constraint. JamesBWatson (talk) 09:33, 5 February 2010 (UTC)

But that's what I'm saying, completely. I'm saying that, if you know the arcsin of x, how you find the arcsin of 1/2x or of 2x. I believe I said above, "If arcsin(x)=y, what's arcsin(2x) in terms of x and y?". When I said "in terms of", I mean in algebraic terms. Since y can have more than one value, arcsin(2x) can have more than one value as well. Timeroot (talk) 01:58, 7 February 2010 (UTC)

Well, it shows that for x rational, arcsin(2x) can be transcendental. But math loses none of its beauty that way. I don't know of a formula for arcsin(2x) (or 3x, etc.). I didn't see one in Abramowitz and Stegun, either. Ozob (talk) 19:01, 10 June 2009 (UTC)

Is there any way to show that if arcsin(x) isn't transcedental, that arcsin(x/2) won't be either? Ow a way to prove that it *can* be transcedental? Timeroot (talk) 05:45, 26 August 2009 (UTC)

I don't know. The only number x I know of for which arcsin(x) isn't transcendental is x = 0. Of course, there are others, but I don't know how to write them down. Ozob (talk) 18:10, 29 August 2009 (UTC)

arcsin(x) is transcendental for all algebraic values of x other than 0 by the Lindemann–Weierstrass theorem which shows that for sin(x). Dmcq (talk) 09:03, 7 February 2010 (UTC)

...but the fact that it is transcendental doesn't mean that given arcsin(x)=y, arcsin(2x) arcsin(x/2 )can't be written using only algebraic functions of x and y. If x is 1/2, then y is pi/6. Clearly y is transcendental, and arcsin(2x) is 3*y. It's a useful theorem, no doubt, but inapplicable to this question. :-( Timeroot (talk) 00:13, 14 February 2010 (UTC)

OK, I added the formulas that I did find to the page. They're not quite what you asked for, though. Ozob (talk) 19:17, 10 June 2009 (UTC)

Well, those are neat... thank you. I think the arctangent one is the prettiest by a good deal. Clearly not what I was hoping for, but I'm aware that such formulas may not exist... and this is still very nice. :-) Timeroot (talk) 13:59, 2 July 2009 (UTC)

General linear combination of cosines

I'm considering adding the identity

$a\cos x+b\cos(x+\alpha )=c\cos(x+\beta ),$

where

$c={\sqrt {a^{2}+b^{2}+2ab\cos \alpha }},\,$

and

$\beta =\arctan \left({\frac {b\sin \alpha }{a+b\cos \alpha }}\right)+{\begin{cases}0&{\text{if }}a+b\cos \alpha \geq 0,\\\pi &{\text{if }}a+b\cos \alpha <0.\end{cases}}$

This is similar to, and derived from, the following already listed identity:

$a\sin x+b\sin(x+\alpha )=c\sin(x+\beta ),$

with same $c$ and $\beta$. I was wondering if there would be any objections to adding it to the article. My derivation was kind of pedestrian thus lengthy so I don't want to include it in the proofs companion article. I was wondering if anyone could check my derivation or offer a source to cite or hint whether this is straightforward thus acceptable without proof. Thanks. 128.138.43.211 (talk) 06:46, 5 September 2009 (UTC)

Without a source it can't be added. Also to me it doesn't look particularly interesting, but of course that is just an opinion. JamesBWatson (talk) 22:10, 9 September 2009 (UTC)

Ambiguity / Error in the Product-to-sum formulas

Consider:

$\sin \theta \cos \varphi ={\sin(\theta +\varphi )+\sin(\theta -\varphi ) \over 2}$

$\cos \theta \sin \varphi ={\sin(\theta +\varphi )-\sin(\theta -\varphi ) \over 2}$

Unless $\theta$ and $\varphi$ are explicitly defined differently than they are now, these two formulas should be the same. —Preceding unsigned comment added by 216.183.146.148 (talk • contribs) 21:29, 17 September 2009 (UTC)

No they shouldn't be the same; in fact I can't even think why anyone would think they should, so I can't explain the error made by this editor. Both these formulas are correct. JamesBWatson (talk) 13:06, 21 September 2009 (UTC)

They both say the same thing; they just have the names of the two variables interchanged. Remember that sine is an odd function so if you say

$\cos \theta \sin \varphi ={\sin(\theta +\varphi )-\sin(\theta -\varphi ) \over 2},$

then that's the same as

$\cos \theta \sin \varphi ={\sin(\theta +\varphi )+\sin(\varphi -\theta ) \over 2}.$

So they really are the same thing. Michael Hardy (talk) 17:51, 20 January 2010 (UTC)

Yes, that is clearly true, but I'm not sure whether that was what the original poster meant. It may have been, but try as I can, I am not able to read his/her words that way. However, at least you have explained why someone might think them "the same". JamesBWatson (talk) 16:23, 25 January 2010 (UTC)

Introduction image

Could we simplify the first image in the introduction by removing those functions that are almost never actually used, like exsec and versin? --Apoc2400 (talk) 11:21, 26 November 2009 (UTC)

I don't see the point. The whole purpose of the diagram is to make the point that all the trig functions can be so represented. Nobody is expected to make more detailed use of the diagram than that, and besides, if anyone does want to look at it in more detail it is not so complicated as to be too confusing. JamesBWatson (talk) 12:24, 26 November 2009 (UTC)

So, what is the point other than to scare off readers? --Apoc2400 (talk) 13:18, 26 November 2009 (UTC)

I'm not sure I really expect 'readers' for this page. It is a list of reference data and the illustration gives reference type data. The articles Trigonometry or Trigonometric functions are more where people read an article. Dmcq (talk) 13:59, 26 November 2009 (UTC)

I suppose the illustration could say something like 'many of these terms are no longer in common use'. Dmcq (talk) 14:02, 26 November 2009 (UTC)

There are also these images, which could be used there:

The sine, tangent, and secant functions of an angle constructed geometrically in terms of a unit circle. The number θ is the length of the curve; thus angles are being measured in radians. The secant and tangent functions rely on a fixed vertical line and the sine function on a moving vertical line. ("Fixed" in this context means not moving as θ changes; "moving" means depending on θ.) Thus, as θ goes from 0 up to a right angle, sin θ goes from 0 to 1, tan θ goes from 0 to ∞, and sec θ goes from 1 to ∞.

The cosine, cotangent, and cosecant functions of an angle θ constructed geometrically in terms of a unit circle. The functions whose names have the prefix co- use horizontal lines where the others use vertical lines.

Michael Hardy (talk) 14:03, 27 November 2009 (UTC)

Yes, those are perfectly alright. However, the existing diagram shows in one simple step the point that all of the functions can be seen as parts of a single diagram. We could have several different diagrams illustrating different groups of functions, but this would involve looking at a string of diagrams, and would not so clearly make the single point "all of them can be taken from one arrangement". I still do not see that the single diagram is too complex for anyone to see whichever function they like, if they look for it. And are we really to believe that anyone looking for information on trigonometric identities is going to be "scared off" because they see a diagram that they don't like? JamesBWatson (talk) 11:11, 3 December 2009 (UTC)

Error in List_of_trigonometric_identities#Double-.2C_triple-.2C_and_half-angle_formulae

quote: As a consequence, it is not possible to express the trigonometric values of angles that are not multiples of 3 degrees divided by any power of two, if using a real-only algebraic expression (for example sin(1°)).

This is not true. You may express cos(360°/17). See 17-gon. Bo Jacoby (talk) 14:54, 7 June 2010 (UTC).

Perhaps they meant exact number of degrees, after all one can always halve an angle even. I'm not sure where they got the real only bit from but it is unnecessary. We really should be sticking in citations for statements though so I'm happy it is deleted. Dmcq (talk) 16:22, 7 June 2010 (UTC)

One can, however, make a correct statement along these lines by saying that the only angles whose sines and cosines can be exactly represented using plus, minus, times, divide, and square root are constructible, i.e., the corresponding point on the unit circle is a constructible number. Such numbers are well-understood. For instance, it's pretty easy to show that for most values of n, an equal division of the circle into n equal parts does not have constructible coordinates, and hence the sine and cosine of 2π/n can't be represented as above. (To be precise, the equal division can be performed if n is a power of two times any number of distinct Fermat primes. A consequence of this is that for all n not equal to a power of 2, there is an angle which cannot be divided into n congruent subangles.)

Unfortunately I don't know of a way to extend this to other operations (cube roots, fifth roots, etc.). Ozob (talk) 01:04, 9 June 2010 (UTC)

For square roots any polygon with 2^a times some distinct Fermat primes is constructible and adding cube roots it is 2^a3^b times some distinct Pierpont prime which are those of the form 2^c3^d+1. However you can construct other angles as well. Dmcq (talk) 11:19, 9 June 2010 (UTC)

cosec rendering

Currently, in my browser (Firefox 3.6.10) in the Pythagorean section I get, in big red letters:

   Failed to parse (unknown function\cosec): 1 + \tan^2\theta = \sec^2\theta\quad\text{and}\quad 1 + \cot^2\theta = \cosec^2\theta.\!

Is this likely to be my browser being non-compliant, or is the formula incorrectly encoded? 175.45.146.134 (talk) 11:38, 26 September 2010 (UTC)

It's not your browser; it's an incorrect edit made by someone else. It's been fixed. Ozob (talk) 12:49, 26 September 2010 (UTC)

New multiple angle formulae

Two new formulae have recently been inserted into the article. Here's the first:

$\sin n\theta =\sum _{k=0 \atop k=1\,{\bmod {\,}}2}^{n}(-1)^{\frac {k-1}{2}}\,{\binom {n}{k}}\,\cos ^{n-k}\theta \,\sin ^{k}\theta$

I'm not sure why this is an improvement. In this formula (for sine), what happens when "k" is two, for example? Apparently we get the square root of negative one on the right-hand side. Why not restore the previous simpler formula that didn't have any complex or imaginary numbers in it? The notation underneath sigma seems somewhat obscure, and it doesn't become more understandable by consulting Modulo. The old formula seemed fairly simple and straightforward: $\sin nx=\sum _{k=0}^{n}{\binom {n}{k}}\cos ^{k}x\,\sin ^{n-k}x\,\sin \left({\frac {1}{2}}(n-k)\pi \right)$

Moreover, the latter (old) formula seems to be the one favored in reliable sources.[2] Anythingyouwant (talk) 16:11, 27 August 2011 (UTC)

When k is two you don't add the term because you only add the terms for odd k i.e. if k is 1 mod 2. It could be written as odd(k) or even(k) for the other one. The formula does not involve any complex numbers but does show the relationship to their use in de Moivre's formula. The trick using sin to generate 0 1 or -1 depending on k is far more obscure and does not match up in an obvious way to de Moivre's formula. It is not straightforward to generate all the terms and then multiply half of them with zero to remove them again. None of them has a reliable source so I'll try and find one as that should clear up the matter, I think what was there was somebody messing around trying to be clever but just obscuring things. I'm very surprised you have not come across the modulo operation, I believed it was done in most schools nowadays and that children would normally know about it before they come across the trig functions at this level. I'll stick in odd and even instead for the moment. Dmcq (talk) 16:49, 27 August 2011 (UTC)

Note that I provided a reliable source at the end of my previous comment.Anythingyouwant (talk) 17:01, 27 August 2011 (UTC)

It's not really but it's as good as any so I'll revert to what was there as the ones I've seen with google just give the first few terms and put etc at the end. Dmcq (talk) 18:03, 27 August 2011 (UTC)

By the way if you look down further at Mathworld you'll see the formula you removed from the deMoivre article. It has no support elsewhere either and what was in the article looked like a copy so I'm happy for it to be gone. Dmcq (talk) 18:06, 27 August 2011 (UTC)

OK, thanks. I've added some text that hopefully makes it clearer: "In each of these two equations, the first parenthesized term is a binomial coefficient, and the final trigonometric function equals one or minus one or zero so that half the entries in each of the sums are removed."Anythingyouwant (talk) 18:47, 27 August 2011 (UTC)

Incompleteness

I've had occasion to look up this web page four times in two days and, despite its massive size, on all four occasions none of the trigonometric identities I was looking for was there. They are: $\cot x+\tan x=2\mathrm {cosec} 2x$, $\cot x-\tan x=2\cot 2x$, $1+i\cot x=2/(1-e^{2ix})$, $\sec x=2e^{ix}/(1+e^{2ix})$, ${\frac {\sin y}{\cos x-\cos y}}=\cot {\frac {1}{2}}(y+x)+\cot {\frac {1}{2}}(y-x)$, $\sec x=1+\sum _{1}^{\infty }{\frac {E_{k}x^{2k}}{(2k)!}}$, ${\frac {1}{2}}\tan {\frac {x}{2}}=\sum _{k=0}^{\infty }{\frac {2^{2k+2}-1}{(2k+2)!}}B_{k+1}x^{2k+1}$, $\sin(15)=({\sqrt {6}}-{\sqrt {2}})/4$ and an integral formula I won't bore you with. OK, so some of those are a bit esoteric, but all but the last one are assumed prior knowledge on pages 2 to 5 of Hardy(1949) "Divergent Sequences".Mollwollfumble (talk) 21:02, 23 October 2011 (UTC)

Archive 1

Product-to-sum and sum-to-product identities

These formulas have a name (a classical name). Unfortunately I don't remember exactly the spelling. The name is (or approximately is) prostospheratic formulas or prostoferatic formulas. I know it is a not commonly used name these days but it would be nice if someone remembers it/ finds it and finds even why they were called that way. I am asking this not only for making an addition to thee article but also for myself. I want to remember the name and learn the neaning/ origin of it. Thanks.  franklin  03:53, 26 January 2010 (UTC)

I suppose that might be mildly interesting, but it seems to me that any such names would be too obscure to be of much encyclopedic interest. JamesBWatson (talk) 08:24, 28 January 2010 (UTC)

• Not what I am asking. I am just asking for the name not inviting to any analysis of its encyclopedic value. By the way, the name (the meaning of the Greek origin) gives some geometric interpretation or some application of the formula. But for the moment I just want to remember the name the rest depends on what exactly the name was.  franklin  11:41, 28 January 2010 (UTC)

I believe Prosthaphaeresis is what you're looking for. Dmcq (talk) 11:46, 28 January 2010 (UTC)

• Thanks so much. I knew the name in Spanish but no way I could have guessed all those h's in the name in English. And it seems to be that the name had EV after all since it already have an article. So, a link from here to that article would be useful.  franklin  11:52, 28 January 2010 (UTC)

Hope someone reads this, I do not really post on wiki often, but wanted to get this info out there (I'm sure I'm breaking some rule somewhere but I thought this was important to point out); for the SINX±SINY sum-to-product rule I have that the SIN and COS are switched in the result, not the ± symbols. The way I read it here suggests that if you have sinx+siny then the 2sin(1/2*x+y)cos(1/2*x-y) and if it is sinx-siny then you get 2sin(1/2*x+y)cos(1/2*x+y), thinking only that the symbol (+ or -) relating the two angles of COS are changed in the result. I'm posing that if sinx+siny then 2sin(1/2*x+y)cos(1/2*x-y) or if SINx-SINy then 2COS(1/2*x+y)SIN(1/2*x-y), only the sin and cos are switched, everything else stays the same. Am I totally off? Info from a calculus MathXL online course (precalc review chapter) problem GR.4.111. I don't know the html tags to make the time i worte this and all that stuff official :( sorry. you can delete this or revise it to fit in better, assuming my comment made any sense (or is even valid). nagromltNagromlt (talk) 21:40, 16 March 2011 (UTC)

I believe that there is an error in these identities, at least the first sum-to-product identity; they should not have both (+) and (-) operators in each function. The cosine function should only have the (-) sign while the sine function should only have the (+) sign. Without demonstrating a proof here is a quicker way using listing of a simple GNU Octave (Matlab) script that clearly shows it. Note that when the operators are reversed it results in a phase shift of the resulting waveform. CODE LISTING :

%GNU Octave script for demonstrating sum to product Identity
%Dan Dady, Feb 2, 2012
clear all; close all

%.............. Create independent variables
t = 0:0.01:100; %first variable
t1 = t.*0.9; %second variable

%.............. Left side of identity .....................
A = sin(t); %first sinusoid
B = sin(t1); %second sinusoid
C = A+B; %superposition

%.............. Right side of identity ....................
D = sin((t+t1)./2); %sine add
E = cos((t-t1)./2); %cosine subtract
F = 2.*(D.*E);

G = sin((t-t1)./2); %sine subtract
H = cos((t+t1)./2); %cosine add
J = 2.*(G.*H);

%.............. Plot waveforms .............................
figure %figure 1 sine add, cosine subtract
plot(t,C,t,F) %note that these waveforms are indeed identical

figure %figure 2 sine subtract, cosine add
plot(t,C,t,J) %note that these waveforms are NOT the same

63.231.234.97 (talk) 17:11, 1 February 2012 (UTC)

Oops, above entry entered in error please disregard since I failed to simultaneously change the operator in the left side of the equation.63.231.234.97 (talk) 17:36, 1 February 2012 (UTC)

Triple tangent/cotangent identities

Perhaps my edit of 04:43, 23 November wasn't clear. I believe that, if it is acceptable to give these things names, the way I have done it is correct. I don't understand the objection that "changing the cotangent identity to the sine-double identity (even if accurate) is not [acceptable]". It wouldn't be accurate, and I never meant to imply such a thing. It is not my intention to give the equation about tangents any name other than the tangent identity, the one about cotangents any name other than the cotangent identity, or the one about sines any name at all. I'm not aware that the one about double sines has any name, though if someone comes up with one, that's fine. SamHB (talk) 16:58, 23 November 2013 (UTC)

Proof of Chebyshev formulae

Just in case it's useful to anyone, the Chebyshev formulae for $\sin nx$ and $\cos nx$ are most easily proved as the imaginary and real parts (respectively) of the formula:

$\operatorname {cis} nx=2\cdot \cos x\cdot \operatorname {cis} ((n-1)x)-\operatorname {cis} ((n-2)x)$

where $\operatorname {cis} x=\cos x+i\sin x=e^{ix},$ as in Euler's formula. Because this is an exponential function, it has the particularly simple summation function $\operatorname {cis} (x+y)=\operatorname {cis} x\operatorname {cis} y$, and $\operatorname {cis} nx=\operatorname {cis} ^{n}x.$

Given this, the formula can be simplified by dividing both sides by $\operatorname {cis} ((n-2)x)$ to get:

$\operatorname {cis} 2x=2\cdot \cos x\cdot \operatorname {cis} x-1.$

Which can be proved by expanding and then simplifying the right-hand side:

${\begin{aligned}2\cdot \cos x\cdot \operatorname {cis} x-1&=2\cdot \cos ^{2}x+2i\cdot \cos x\cdot \sin x-\cos ^{2}x-\sin ^{2}x\\&=\cos ^{2}x+2i\cdot \cos x\cdot \sin x-\sin ^{2}x\\&=(\cos x+i\cdot \sin x)^{2}\\&=\operatorname {cis} ^{2}x\\&=\operatorname {cis} 2x\end{aligned}}$

71.41.210.146 (talk) 17:06, 28 December 2013 (UTC)

Symmetry

Forgive me if I'm being an idiot, but in the table that includes the cofunction identities, should the transformations not be described as reflections in $\theta =0,\theta ={\pi \over 4}$ and $\theta ={\pi \over 2}$ respectively? M.A.Redman (talk) 19:14, 10 April 2014 (UTC)

useful identity

It would be handy to give a solution to the geodesic eqn $a\sin x+b\cos x=c$ 67.198.37.16 (talk) 20:01, 30 September 2015 (UTC)

More than two sinusoids

As in the "Arbitrary phase shift" and "Sine and cosine" cases the expressions for a and θ should be unambiguous. To the best of my knowledge that is

$a={\sqrt {\sum _{i,j}a_{i}a_{j}\cos(\theta _{i}-\theta _{j})}}$

and

$\theta =\operatorname {atan2} \left(\sum _{i}a_{i}\sin \theta _{i},\sum _{i}a_{i}\cos \theta _{i}\right).$

— Preceding unsigned comment added by Tpreclik (talk • contribs) 03:39, 3 January 2017 (UTC)

Pythagorean Identity

When I learned the following identities, they were all called "Pythagorean identities" with consistency from teacher to teacher:

$\sin ^{2}x+\cos ^{2}x=1$

$\tan ^{2}x+1=sec^{2}x$

$1+\cot ^{2}x=\csc ^{2}x$

However, online it seems that some people only use the term "Pythagorean identity" to refer to the former. Therefore, would anybody object to me changing "identity" to "identities"? Thank you.LakeKayak (talk) 19:18, 18 February 2017 (UTC)

As it seems that nobody has an opinion, I am going to make the change and wait for people's responses from there.LakeKayak (talk) 01:28, 21 February 2017 (UTC)

If the article gets too long, how shall we reorganize or split it?

By WP:SIZESPLIT and Wikipedia:Article size pages above 60 kb size should be split List of trigonometric identities is about 71 kb so is in need of a split.

Also the list is ab bit hard to follow so lets have a discussion about how we reorganize this page and maybe if and how how we should split it.

My idea is to split it in two, one with let us call it the high school trigonometry and a second one with more advanced subjects.

Advanced subjects would for me include:

• List of trigonometric identities#Historical shorthands maybe better at a page historical trigonometric functions
• List of trigonometric identities#Angle sum and difference identities matrix forms
  • Sines and cosines of sums of infinitely many terms
  • Tangents of sums
  • Secants and cosecants of sums
• List of trigonometric identities#Multiple-angle formulae
  • Chebyshev polynomial
  • Hermite's cotangent identity

and many more lets discuss before we edit that makes nobody happy. WillemienH (talk) 12:44, 25 February 2016 (UTC)

I think the article is on a coherent single topic and so should not be split up. I don't think it's hard to navigate, since there's a table of contents. With two separate articles, someone might have a hard time finding what they're looking for, if they go to the wrong article. Loraof (talk) 20:31, 5 June 2016 (UTC)

I agree. I don't think splitting up is a good idea. The cited guidelines with the 60kb are more than a crude rule of thumb than an absolute figure. We have plenty of compilation and overview articles that are larger than 60kb and where splitting them up is simply not a good option. In fact many of our excellent articles are > 70kb.--Kmhkmh (talk) 20:53, 5 June 2016 (UTC)

I don't know if splitting is needed but holy crap is this page difficult to read with the current format and organization. — Preceding unsigned comment added by Bofum (talk • contribs) 04:44, 1 October 2018 (UTC)

Significant identities

There are a great many existing trigonometric identities. However, this article cannot contain all of them and has space only for the more significant identities. Please do not insert identities that are not in common use and especially do not include a huge list of them.—Anita5192 (talk) 17:49, 17 November 2019 (UTC)

I wholeheartedly agree and would go even further. There are many identities on this page that are not sourced or sourced to unreliable sources. The page seems to be a magnet for people who want to see "their" identities in print and needs to be pruned from time to time to weed these things out. I've also noticed some basic identities (that should be in the list) without citations, so a straightforward weeding would have to be done with care and is probably a job for more than one editor.--Bill Cherowitzo (talk) 23:08, 17 November 2019 (UTC)