Talk:List of trigonometric identities

If the article gets too long, how shall we reorganize or split it?
By WP:SIZESPLIT and 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 maybe better at a page historical trigonometric functions
 * List of trigonometric identities matrix forms
 * Sines and cosines of sums of infinitely many terms
 * Tangents of sums
 * Secants and cosecants of sums
 * List of trigonometric identities
 * 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)

atan2
I recommend removing all mention of atan2 from the article, as atan2 is a computer programming language function—not a standard trigonometric function.—Anita5192 (talk) 16:35, 10 November 2019 (UTC)

I support that. There are only two sections where atan2 is mentioned: So neither of these uses of atan2 are actually supported by the sources, at least that I can verify. The first one can be removed without any fuss. The second one can be rewritten in terms of standard arctan based on the Mathworld article. -Apocheir (talk) 23:50, 10 November 2019 (UTC)
 * There's an identity for atan2 in the table under List_of_trigonometric_identities that is equivalent to the one given for arctan, and is also stated in Atan2. It's not cited here, although a proof is given there.
 * The other section is List_of_trigonometric_identities. Looking at the sources, the Mathworld one doesn't actually use atan2, the Cazelais source is a broken link, and I can't verify the Apostol source but it's unlikely that it uses atan2. (It's dated 1967, while atan2 was first introduced to Fortran in 1961, and I doubt the terminology moved into a standard calculus text that early.)

Sure, atan2 originates from programming languages, but it is nevertheless a perfectly valid mathematical function. It would be useful for the linear combination section, because


 * $$a\cos x+b\sin x = c\cos(x+\varphi)$$

can be made to work for all a and b by using


 * $$c = \sqrt{a^2 + b^2},$$


 * $$\varphi = \operatorname{atan2}(-b, a) = \operatorname{arg}(a - bi),$$

with the interpretation that


 * $$0 \cdot \textrm{undefined} = 0.$$

The way it is now, using atan, leaves an unnecessary singularity at a = 0, and the equations do not hold there. The phase range is also only 180 degrees instead of 360, and amplitudes can take on negative values, which is a bit strange. If it's the programming language origin of atan2 that's the main issue, then you can equivalently use the arg function as above.

Perhaps one of the most common places for this linear combination to occur is in the Fourier series, where one converts


 * $$a_n \cos\left(\tfrac{2\pi }{P}nx\right) + b_n \sin\left(\tfrac{2\pi }{P}nx \right)$$

into


 * $$A_n\cdot \cos\left(\tfrac{2\pi }{P}nx - \varphi_n \right).$$

That article defines


 * $$A_n \triangleq \sqrt{a_n^2+b_n^2}$$ and $$\varphi_n \triangleq \operatorname{arctan2}(b_n,a_n),$$

unabashedly using the atan2 function. In this prototypical application it would be strange to use a 180 degree phase range and negative amplitudes, which is part of the reason why such a parametrization feels weird to me. More generally, a full circle is the most natural range for angles, and amplitudes are most natural if they're non-negative.

I don't unfortunately have suitable sources at hand, so I'm leaving the article as-is, but if someone finds such then I suggest using arctan2 in the article. You can also use arg, but there's no need to introduce a complex function into a real context, no matter how suited they are for expressing things related to the unit circle. -- StackMoreLayers (talk) 02:13, 12 April 2021 (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)

CPU Usage
This page, despite seeming to have no active content, utilizes 100% of the core the thread is running on. Google Crome Version 84.0.4147.105 (Official Build) (64-bit) — Preceding unsigned comment added by 75.109.252.140 (talk) 17:11, 4 August 2020 (UTC)

Overview figure
User created a new overview figure, and replaced  the original with the new version in the article:

Pending discussion and consensus, I have restored the original diagram, as the new one seems entirely unreadable without clicking on it to open the full-size source. Comments welcome. - DVdm (talk) 17:35, 22 September 2021 (UTC)

Note: same on articles Trigonometric functions and Unit circle. - DVdm (talk) 17:39, 22 September 2021 (UTC)


 * I created the new diagram because the original one contained far too many distractions (see chartjunk). A couple years ago I consulted this diagram because I was asked to memorize the values in a class. The three different text colors, varying font sizes, and varying text directions make the initial diagram very hard to read. I can enlarge the font size in the new SVG file if readers prefer that. Crossover1370  (talk &#124; contribs) 18:11, 22 September 2021 (UTC)


 * Good idea, you can create another version with different font and/or size and add to the galery here, so we can compare and try to agree. - DVdm (talk) 18:27, 22 September 2021 (UTC)


 * I don't see what distractions you are referring to, since both diagrams contain the exact same information. The only difference is the colors, which in my opinion make the original diagram more readable: multiples of π/6 are blue; those of π/4, red.—Anita5192 (talk) 18:33, 22 September 2021 (UTC)
 * Indeed. And those of π/2, black. That's exactly what I was thinking too. - DVdm (talk) 18:41, 22 September 2021 (UTC)


 * Apart from the color, there are also the varying font sizes and the changing direction of text. It is far easier to read text that is upright than text that is rotated 30, 45, or even 60 degrees. I actually believe that the rotating text is the biggest issue with the readability of the original diagram. Crossover1370  (talk &#124; contribs) 19:16, 22 September 2021 (UTC)


 * Black text laid over a black line is very hard to read: see in the new image how the leg of the π in 4π/3 runs together with the line. Simplifying a figure does not necessarily improve it: the different colors and angles used in the old image enhance understanding, rather than take away from it. -Apocheir (talk) 20:34, 22 September 2021 (UTC)
 * I am willing to increase the size of the transparent zone or even change the text in the latter image to blue to increase its readability. I am not a fan of excessive use of color such as in the first image. However, I stated earlier that the biggest problem with the old image is not the color but rather the changing text direction - how much of the text is rotated 30°, 45°, or even 60°. Such text is much harder to read than upright text. Crossover1370  (talk &#124; contribs) 21:24, 22 September 2021 (UTC)


 * I don't have any problem reading the non-horizontal text. Indeed, years ago, in the days of manual drafting, we occasionally used non-horizontal dimensions to fit the diagram.—Anita5192 (talk) 02:37, 23 September 2021 (UTC)
 * Sure, it is not hard to read the coordinates (I can clearly see that the coordinates at π/3 are (1/2, √$\overline{3}$/2)). But for students who are trying to memorize the diagram, the nearly-vertical text in some places makes the diagram as a whole harder to read (some may have to tilt their heads) and harder to memorize. Crossover1370  (talk &#124; contribs) 03:41, 23 September 2021 (UTC)
 * I think that memorizing such diagrams is a not a good idea. In my experience, in order to recall access to a sin/cos-value, the only thing to memorize, was the trivial recipe to reproduce the table in List of trigonometric identities and to understand how to mirror things to the other quadrants . - DVdm (talk) 08:33, 23 September 2021 (UTC)

new suggestion
In connection with Product-to-sum and sum-to-product identities,how about adding this article:https://wikimedia.org/api/rest_v1/media/math/render/svg/2073f2b5e451b6da9a646016cd076041c565f6bb(from https://fr.wikipedia.org/wiki/Identit%C3%A9_trigonom%C3%A9trique)? This is Tangent formura of it.And using Negative angle formula,I can show like this. $$\tan\theta\pm\tan\varphi=\frac{\sin(\theta\pm \varphi)}{\cos\theta\cos\varphi}$$240D:1E:309:5F00:5A:9595:FDF:A1F7 (talk) 06:50, 11 December 2021 (UTC)
 * I added it to the table. Danstronger (talk) 14:07, 11 December 2021 (UTC)

Incomplete list misses trivial identities
I think $$\tan(\theta) = \frac{sin(\theta)}{\cos(\theta)}$$ but I'm not sure, since it's not listed on Wikipedia. John Moser (talk) 23:43, 29 January 2022 (UTC)Here is a citation.Could you edit the deutsch version?I changed　my computer model.I change citation. https://www.doubtnut.com/question-answer/in-a-triangleabc-prove-that-cos22a-cos22b-cos22c1-cos2acos2bcos2c-13104　,and because sin^2(x)+cos^2(x)=1,it become like this. --240D:1E:309:5F00:F6F7:FABA:AD23:F5C6 (talk) 05:32, 1 April 2022 (UTC)

Suggestion for improving subsection on Ptolemy's theorem
This subsection needs improvement, I think: [|"Ptolemy's theorem"]

There are a few problems with it.

First, although Ptolemy's theorem does indeed relate nicely to the sum and difference trig identities, that relationship needs one of the quadrilateral's diagonals to pass through the circle center, i.e. to be a diameter. For example, here: Sine, Cosine, and Ptolemy's Theorem.

The current wikipedia page section does not refer to that relationship, or say anything about one diagonal being a diameter.

Second, a figure would really help a lot. I would be happy to provide one, but I wanted to run this by the talk page first, before I start going in and deleting existing content.

Third, the current explanations offered are not really very explanatory. The word "trivial" is asserted three times, but this doesn't really help the reader to follow what the logical flow is meant to be. Then, after the three trivial-statements, a fourth statement suddenly emerges, but it really isn't clear how it follows from the previous ones. For example, the fourth line presumably is meant to follow from the third line, but the fourth line includes mention of the variable y, which doesn't feature in the third line at all. Although the statement in the fourth line is indeed true, that doesn't mean that it logically follows from the preceding statements. If it does follow, then the relationship is unclear at best.

I suggest (and would be happy to implement) replacing this subsection with something more closely modeled on the Cut The Knot page linked above. The Maor book "Trigonometric Delights" that is referenced by that page could also serve as the ref for this section of the wiki page. (The relevant pages from that book are pp.93-94 of the 2020 printing). — Preceding unsigned comment added by RajRaizada (talk • contribs) 15:48, 25 May 2022 (UTC) RajRaizada (talk) 20:00, 25 May 2022 (UTC)


 * Since nobody seemed to object to my suggestion from a couple of weeks ago of editing the section on Ptolemy's theorem, I made those proposed changes just now. I hope the edits strike people as being an improvement. RajRaizada (talk) 21:52, 13 June 2022 (UTC)
 * The section now doesn't address the trig identity aspect at all, though. It's just covering the same material as Ptolemy's theorem. If it's not about the trig identity, it shouldn't be on this page.
 * I have doubts if the trig identity aspect of Ptolemy's theorem should be here at all. It's not all that useful. Apocheir (talk) 23:40, 13 June 2022 (UTC)
 * These are interesting points. I agree that deriving the sum and difference formulas from Ptolemy's theorem does seem a bit roundabout. I looked into this a bit, and it appears that the relation to this theorem is important historically, as it is how these sum and difference trig formulas were first derived. E.g. see ref 19 on this wikipedia page: History of trigonometry
 * I added a couple of sentences, and a link to the History of trig wikipedia page, to the trig identities page, in order to make this connection explicit. RajRaizada (talk) RajRaizada (talk) 01:10, 14 June 2022 (UTC)
 * It struck me that another reason why the Ptolemy's theorem section doesn't quite seem to belong is that whoever first added it put in the wrong section. That theorem gives rise to the sum and difference formulas, but it had been placed as a subpart of the section "Product-to-sum and sum-to-product identities". I had been so preoccupied by previous version's excessive use of the assertion "trivial" and its failure to include a figure that I hadn't initially noticed that!
 * I moved the Ptolemy's theorem section just now so that it is forms the final subpart of the section "Angle sum and difference identities". I hope this change strikes people as an improvement. RajRaizada (talk) 16:21, 14 June 2022 (UTC)

Should these new identities be added?
These identities are special because this one is about constructing regular polygons and the minimal polynomial for 2cos(2π/n).

$$6\cos\left(\frac{2\pi}{7}\right) = 2\sqrt{7}\cos\left(\frac{1}{3}\arccos\left(\frac{\sqrt{7}}{14}\right)\right) - 1$$

$$12\cos\left(\frac{2\pi}{13}\right) = \sqrt{104 - 8\sqrt{13}}\cos\left(\frac{1}{3}\arctan\left(\frac{\sqrt{3}(\sqrt{13} - 1)}{7 - \sqrt{13}}\right)\right) + \sqrt{13} - 1$$

$$P(x) = \sum_{n = 0}^{k}(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{k - \left\lfloor\frac{n + 1}{2}\right\rfloor}{\left\lfloor\frac{n}{2}\right\rfloor}x^{k-n}$$

$$\Rightarrow P\left(2\cos\left(\frac{2\pi}{2k + 1}\right)\right) = 0$$

Does anyone agree that they should be added? 173 Ascension 257 (talk) 18:31, 28 December 2022 (UTC)


 * What's the source? Apocheir (talk) 01:39, 29 December 2022 (UTC)
 * The source of the first two identities is made by American mathematician Andrew M. Gleason, who has published these identities, from:
 * http://math.fau.edu/yiu/PSRM2015/yiu/New%20Folder%20(4)/Downloaded%20Papers/AMMGleason1988.pdf
 * The polynomial was something else, which is from:
 * https://en.wikipedia.org/w/index.php?title=Minimal_polynomial_of_2cos(2pi/n)&oldid=1070293024#cite_ref-3
 * The sequence that I found is from:
 * https://oeis.org/A187360/a187360.pdf
 * ...and the cases for n = 1 to 30 seem to match what the series gives.
 * I could hardly find the exact polynomial from any other source, yet these sources are related to the series that gives the minimal polynomial of $$2\cos\left(\frac{2\pi}{2k + 1}\right)$$, or $$2\cos\left(\frac{2\pi}{n}\right)$$ for all positive odd numbers n, yet these related articles may help:
 * https://www.jstor.org/stable/2324301?read-now=1&seq=1#page_scan_tab_contents
 * https://www.jstor.org/stable/2301023?read-now=1&seq=1#page_scan_tab_contents 173 Ascension 257 (talk) 23:12, 30 December 2022 (UTC)

Does infinite sum or product expansion of trig functions belong here?
Discussion is opened as per recommendation of user @Apocheir. Let us give some time for this to be discussed here before choosing to remove content. I have added the section with citation so there is no harm done in the time being. EditingPencil (talk) 21:09, 30 November 2023 (UTC)


 * Mostly my issue was seeing yet another uncited identity (albeit a familiar one) added to the page. That said, I question whether either of these are useful as trigonometric identities. As derivations of the relationship between trig functions and the exponential function, or as approximations of the functions, sure, but that's not what this page is about.
 * The Taylor expansion of sine and cosine is covered adequately on Sine and cosine. The product formulas, well, maybe I'm not well enough versed in special functions to immediately see what their point are, but they don't strike me as very useful. This page is so long that Who Wrote That? chokes before it can tell me who added it and when.
 * The goal of this page is not to repeat every identity in Abramowitz and Stegun. (That might be a good WikiSource or WikiBook project, though.) Apocheir (talk) 21:59, 30 November 2023 (UTC)
 * Fair but I disagree. It takes up very little space and it is very useful when you have to express trig functions in the n-th order and those formula can be used to prove things as well. In fact, those two trig functions can alternatively be defined as such to begin with, like otoh, Andrew Gleason does in his book on Abstract analysis. I didn't initially bother citing them because of how often I come across them as a student in physics. The product formula is less useful in comparison, although my knowledge of this is also very limited. EditingPencil (talk) 23:20, 30 November 2023 (UTC)
 * @Apocheir Let me know if this resolves the issue. Otherwise, I recommend we give time for others to comment before making changes. Those 5 extra lines are not really an urgent matter and they're even helpful imo.
 * PS: I reverted one of your edits due to wrong use of terminology, I also recommend you discuss that here before trying to merge the sections. Thanks for understanding! ^^ EditingPencil (talk) 23:30, 30 November 2023 (UTC)
 * Making the change first, and then finding out if anyone disagrees with it, is the standard practice on Wikipedia. Read WP:BRD. Apocheir (talk) 23:52, 30 November 2023 (UTC)
 * I meant you can discuss if you are planning to undo reversion. EditingPencil (talk) 00:09, 1 December 2023 (UTC)
 * I think we should add an article called trigonometric identity which consists predominantly of prose (rather than formulas) and discusses the most important identities along with general material about the context (why were identities historically important, why do they appear more in trigonometry than other subjects, how are they used in education, etc.), and then we can leave this article as a sink to park arbitrarily much stuff that people can find sources for. Anything especially important should also have its own dedicated article.
 * In general "list" articles aren't ever very legible, and I don't think this one can really be put into any great kind of organization. If it ends up reproducing every relevant bit of material in Abramowitz and Stegun, that seems fine. That's still useful as a reference. –jacobolus (t) 00:47, 1 December 2023 (UTC)


 * I think we should remove these infinite sums and products for the reason that they are not really trigomentric identities, which should i.m.o. have at least two trigonometric functions in the equation. - DVdm (talk) 23:56, 30 November 2023 (UTC)
 * Then that is also what the article must state at the beginning. Instead, these equations are fine under that description. I think removing these is unnecessary. I think the discussion should also include whether this section can be pushed off into 'miscellaneous'. EditingPencil (talk) 00:14, 1 December 2023 (UTC)
 * btw proofs on trigonometric identities has few extra identities which have only one trig function but I wont edit them here since they can be trivially derived from infinite series expansion. EditingPencil (talk) 00:23, 1 December 2023 (UTC)
 * Proofs of trigonometric identities is a terrible scope for an article. There's no tight unifying theme, there's no obvious narrative organization, and the possible scope is almost unbounded. The current version is also entirely unsourced, and some may be original research.
 * That article should be split apart with proofs moved to dedicated articles about each separate identity if they are relevant. Some of it can just be deleted; not every math proof is particularly encyclopedic. Some instead belongs in a textbook or the solutions manual of a problem workbook. –jacobolus (t) 00:53, 1 December 2023 (UTC)