Talk:List of trigonometric identities

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:

• There's an identity for atan2 in the table under List_of_trigonometric_identities#Angle_sum_and_difference_identities that is equivalent to the one given for arctan, and is also stated in Atan2#Angle_sum_and_difference_identity. It's not cited here, although a proof is given there.
• The other section is List_of_trigonometric_identities#Linear_combinations. 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.)

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)

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 Crossover1370 (talk · contribs) created a new overview figure, and replaced ([1]) the original with the new version in the article:

• 
  (original)
• 
  (new)

Pending discussion and consensus, I have restored ([2]) 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 ([3]) and Unit circle ([4]). - 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 | 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)

@Crossover1370: 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 | 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 | 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, √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 | 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#A useful mnemonic for certain values of sines and cosines 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)

Half-angle formulas

I added the following half-angle formulas: $${\begin{aligned}\sin {\frac {\theta }{2}}&={\frac {\sin \theta }{\sqrt {2(1+\cos \theta )}}},\quad |\theta |<\pi \\[3pt]\cos {\frac {\theta }{2}}&={\frac {\sin \theta }{\sqrt {2(1-\cos \theta )}}},\quad 0<\theta <2\pi \\[3pt]\sec {\frac {\theta }{2}}&={\frac {\sqrt {2(1-\cos \theta )}}{\sin \theta }},\quad 0<\theta <2\pi \\[3pt]\csc {\frac {\theta }{2}}&={\frac {\sqrt {2(1+\cos \theta )}}{\sin \theta }},\quad |\theta |<\pi \\[3pt]\end{aligned}}$$ Are they correct? Eric Kvaalen (talk) 12:19, 30 September 2024 (UTC)

I reverted your edit per policy WP:PROVEIT. It doesn't matter whether they are correct. You need sources for them, if only to demonstrate notability. This is correct too:

$\sin {\frac {\theta }{2}}+\sin {\frac {\theta }{2}}=2\sin {\frac {\theta }{2}},$

but we don't add it to the aticle because it's only notable when a relevant reliable source mentions it. This was amply explained to you before:- see User_talk:Eric_Kvaalen#Original_research - DVdm (talk) 13:46, 30 September 2024 (UTC)

Ah, yes, you're the one who wouldn't let me put how much the planets pull on the earth. And you said something on my talk page about how you spend your time preventing people looking for interesting things on Wikipedia. I didn't understand that. So do we need a reference for everything that is said in Wikipedia? Even something like ${\frac {\sin \theta }{{\sqrt {2(1+\cos \theta )}}{\sqrt {(1-\cos \theta )/2}}}}=1$ when $\sin \theta >0?$ Eric Kvaalen (talk) 11:09, 1 October 2024 (UTC)

No, I said something on your talk page about how I you spend some of my time preventing people finding uninteresting things on Wikipedia.

Yes, even, and specially something like that. See wp:BURDEN. If you don't like it, there are other places. - DVdm (talk) 11:55, 1 October 2024 (UTC)

Well, you wrote, "I spend some time preventing others to spend a lot of time looking for things that people have put into Wikipedia because they are interesting..." Whatever. Anyway, why do you spend your time enforcing your idea of what is interesting? And what is the point of requiring references for things like this? Eric Kvaalen (talk) 15:08, 1 October 2024 (UTC)

Just read the policies. I spend some of my time guarding them. I will repeat in bold: the point of requiring references for things like this, is to demonstrate notability. If you indeed find this impossible to understand, then I can't help you. - DVdm (talk) 15:15, 1 October 2024 (UTC)

@Eric Kvaalen: The point of this page is not to exhaustively list every possible trigonometric identity that any Wikipedian can make up, which would be overwhelmingly large and not very useful to readers, but rather to organize a list of the most important and influential trigonometric identities, as identified by the best secondary sources we can find. Unfortunately this page doesn't currently live up to its ideal, and there are identities missing which should be listed, identities mentioned which should be elaborated, pictured, or even turned into dedicated articles, and also identities listed which should be cut or removed to more specific articles. The current content of this page is mostly (all?) supportable by reliable sources but not all of it is currently well sourced – that should also be improved. But if you want to add new identities that nobody has yet thought to include in the past two decades since this page was created, you need to provide a source demonstrating that the identity is considered important. Even with sources, we can have discussion here about whether each identity is really pulling its weight, and might decide to trim some material for length. If you discover a new identity that is not included in reliable secondary sources, the appropriate place to publish it is in a peer-reviewed paper (or a preprint, or a blog post, etc.), not here. –jacobolus (t) 15:45, 1 October 2024 (UTC)

Indeed, that's why I also undid this and this. - DVdm (talk) 16:02, 1 October 2024 (UTC)

@Jacobolus: I'm not trying to list every possible identity! The one in the article for $\sin \theta /2$ is inaccurate on a computer when θ is small, because you have subtraction of two similarly sized numbers (1 and $\cos \theta$). The same problem occurs for the others (cos, sec, and csc), either for small θ or for θ near π. This is all very basic computer science and I'm sure it's in the literature, since it's important. Instead of destructively deleting, editors should help by finding references, if they really think it's necessary to prove that the alternative formulas are better in certain cases, even though it's fairly obvious. Eric Kvaalen (talk) 19:38, 1 October 2024 (UTC)

If it is in the literature and is important, then you should find a source, which shouldn't be hard. –jacobolus (t) 19:44, 1 October 2024 (UTC)

Note that ⁠${\tfrac {1}{2}}(1-\cos \theta )$⁠ is the haversine. There are several ways you can compute it accurately if you need a formula that works for small angles; what kind of formula would be useful would depend on the context. (The sine half-angle formula you listed here is a fine enough choice.) The cosine, secant, and cosecant of a half angle don't have much use IMO. –jacobolus (t) 20:22, 1 October 2024 (UTC)

@Jacobolus: Well, the best way to calculate haversine of a small θ is $(\sin(\theta /2))^{2}$. But our problems is to calculate $\sin(\theta /2),$ so that doesn't help. And we want to find it from trigonometric functions of θ, not from θ itself. In other words, in applications we often have the sine and cosine of θ, and we want $\sin(\theta /2).$ We don't want to find θ first and then simply divide by 2 and take the sine. Eric Kvaalen (talk) 19:53, 2 October 2024 (UTC)

@Jacobolus and Eric Kvaalen: it's best to go on about these technicalities on one of your user talk pages, as, per wp:talk page guidelines, we cannot discuss them here in article talk space. We can have discussions about the existence (or quality) of sources, but, unless referring to sources, not about best ways of calculating anything - DVdm (talk) 08:42, 3 October 2024 (UTC)

@DVdm This seems like an unhelpfully hostile response to a quite ordinary Wikipedia talk page discussion. The need for sources has already been mentioned multiple times. –jacobolus (t) 13:20, 3 October 2024 (UTC)

Wasn't meant to be hostile, sorry about that. Just as a reminder of the guidelines. Cheers. - DVdm (talk) 15:50, 3 October 2024 (UTC)

(No worries, and maybe "hostile" is stronger descriptor than I quite meant. But "who would use this identity in practice in what context" seems like an entirely reasonable talk page question to help decide what belongs on a page like this. I agree with you anything added needs a source, and I expect Eric Kvaalen can find one if they go search for it, at least for the half-angle sine identity.) –jacobolus (t) 16:43, 3 October 2024 (UTC)

I think a section on identities used to improve numerical accuracy near certain values would be a good addition, if referenced to the literature.

Doesn't the half-angle cos identity that was removed have the same problem as the half-angle sin identity in the article? Maybe even worse since it's dividing a small quantity by a small quantity near 0 and π... Apocheir (talk) 22:09, 3 October 2024 (UTC)

There's not much issue for cosine because cosine of a small angle approaches 1, so numerically you eventually just bump into 1 and run out of significant digits; the relative error inevitably goes to 0 as your angle gets smaller. (The formula given at the top of this section approaches 0/0, but I haven't tried to figure out how it behaves numerically in approximate arithmetic.) There's a much bigger issue for the versine (⁠$\theta \mapsto 1-\cos \theta$⁠), sine, tangent, chord, etc. –jacobolus (t) 22:26, 3 October 2024 (UTC)

@Apocheir and Jacobolus: Yeah, there is a problem with "my" cos(θ/2) formula around θ=0 (even though it goes toward 0/0, because the denominator is inaccurate), but it avoids that problem around θ=π, whereas the formula in our article has the problem there. That's why both formulae should be given. Why do we need references for things like this which are easily worked out? This is math, not history! Eric Kvaalen (talk) 10:25, 4 October 2024 (UTC)

Because it is way beyond wp:CALC. We do need references for things like this, to show that they are sufficiently interesting for an encyclopedia. That is by design. - DVdm (talk) 11:28, 4 October 2024 (UTC)

The point of Wikipedia is to describe and summarize the content of existing literature, not to present novel results. See Wikipedia:No original research. –jacobolus (t) 14:06, 4 October 2024 (UTC)

@DVdm and Jacobolus: I looked at the link that DVdm provided, and this is what I found: "Wikipedia has no firm rules"! Aren't we able to decide whether something is interesting? I think it's insulting to us Wikipedia editors to say that we can't decide for ourselves whether something is interesting. I had a big argument a while ago when I pointed out that the dates of Easter repeat after a certain time. The others said I had to prove that that is interesting by finding a reference! I think that's ridiculous. And it doesn't take "research" to show that the equations I added here are not only correct but more accurate in certain areas. It's insulting to say that Wikipedia editors are not capable of such simple mathematics and they have to find it written in some book! I only have two math books, one on engineering mathematics, and one on abstract algebra. Ah, plus another on a rather arcane subject. So none of my books will have these simple formulas. Eric Kvaalen (talk) 21:20, 5 October 2024 (UTC)

@Eric Kvaalen "Wikipedia has no firm rules" doesn't mean the same as "intentionally do the opposite of several longstanding Wikipedia policies whenever you personally feel like it, for no particular reason". If you want to push on this one, take it up at e.g. Wikipedia:Village Pump (policy) or Wikipedia talk:No original research.

But the most important and fundamental of all Wikipedia rules is that Wikipedia content is decided by consensus, both on individual pages and site-wide. You certainly haven't demonstrated any local consensus here for adding Wikipedians' new unpublished research into trigonometric identities. –jacobolus (t) 21:29, 5 October 2024 (UTC)

Ditto. - DVdm (talk) 21:47, 5 October 2024 (UTC)

@DVdm and Jacobolus: What I don't understand is what motivates you to enforce the rules (as you interpret them). Nobody requires you to do that. Do you really think the formulas I added are incorrect? No. Do you really think they are useless? No. Do you really not understand that they are better in certain cases? No, you do understand that. Do you really think that readers will find those formulas extremely boring and will think the article is worse if they are there? No, I don't believe you think that. Eric Kvaalen (talk) 12:22, 6 October 2024 (UTC)

Even with a source I would be hesitant to add this formula to this article, because while I can trivially derive it myself, having now done about 20 minutes of literature search I couldn't find it anywhere, which indicates that it's really not widely known/used, and probably doesn't belong on a generic page like this. With a source, I think it would be more appropriate to include somewhere in a dedicated article half-angle formulas (currently a redirect). I think this topic should be expanded into a separate article either way, because List of trigonometric identities does not have the space for description of the context, proofs, images, history, applications, etc.

To answer your other questions: I do really think 3/4 of these formulas are pretty useless and the fourth one has an extremely niche use which I'm having trouble imagining a clear concrete situation for, but I have myself used similar (novel, niche) formulas of this type in the past, in various one-off situations, so I'm sure someone, somewhere could come up with a use for it. That's not good enough for inclusion into Wikipedia though. I can personally think of / have myself derived several dozen more trigonometric identities which are not included on this page which I think are more useful than this one, some of which can even be found in (obscure) sources. If I had been carefully keeping a list of every moderately interesting trigonometric identity I ever saw in a published paper or book somewhere, that would be many hundreds more formulas, again mostly more useful/interesting than these. But adding them here would not really be net helpful; it balloons the page for something that isn't really relevant to readers, which has the effect of hiding what they might be looking for in the noise.

I think you misunderstand Wikipedia's nature. Wikipedia is an encyclopedia, a tertiary source, not a primary source. Its purpose is to organize what reliable secondary sources have already reported/discussed, not to be a venue for new research results. If you publish your formula somewhere, and it proves to be influential, then it might naturally wind up in Wikipedia a few years down the line.

Please read Wikipedia:What Wikipedia is not § Wikipedia is not a publisher of original thought and Wikipedia:No original research. Again though, this isn't really the venue for arguing bedrock Wikipedia policies which have been the solid consensus of editors for 20 years. –jacobolus (t) 15:27, 6 October 2024 (UTC)

@Eric Kvaalen: what motivates me to do anything is  Off-topic on an article talk page. You can ask me on my user talk page or on yours. - DVdm (talk) 16:20, 6 October 2024 (UTC)

Well, DVdm, it's not to me that you need to give account for your deeds. I am not Jesus Christ. II Cor 5:10

@Jacobolus: I don't think an encyclopedia has to be a "tertiary source". Diderot said the Encyclopédie was "to change the way people think" and for people to be able to inform themselves and to know things (Encyclopedia). I strongly disagree with the idea that we editors cannot use our brains enough to make obviously true inferences or to point out things, without having a reference. A while back I wanted the mass of nitrogen in the earth's atmosphere, and I went to Atmosphere of Earth to find out how much nitrogen there is in the air, in mass percent. The number wasn't there, so I did a quick calculation of the mass-based composition of air, from the molar composition, and added it to the article. But a certain guy (with whom I had had arguments previously) reverted it, saying I didn't have a reference! Look, I have tried to argue this in other venues. Please read Wikipedia:Village pump (policy)/Archive 152#Are policies being used to the detriment of Wikipedia? and then Wikipedia:Village pump (policy)/Archive 156#Are policies being used to the detriment of Wikipedia?. That was a discussion in 2020 but it was closed down by an administrator who didn't agree with me, even though there were more people who did agree with me.

I think we should have the right (and I think we do have) to make this article better by adding explanation of why certain formulae should be used rather than others. If you think there are too many formulae, you should be allowed to use your head and decide what should and shouldn't be here. The idea that "Wikipedia is not a publisher of original thought" should not be used to prevent editors from thinking at all! It's amazing what gets called "original research"! Such as calculating the mass-percentages of gases in air.

I'm curious what kinds of trigonometric identities you have come up with, and why!

Eric Kvaalen (talk) 08:53, 9 October 2024 (UTC)

 Off-topic. See wp:talk page guidelines. Warning for abuse on your user talk page. - DVdm (talk) 09:05, 9 October 2024 (UTC)

"I strongly disagree with the idea ..." – That's fine, but Wikipedia runs on consensus, not what one editor personally prefers. You should take this up elsewhere because it is off topic here, and Wikipedia talk pages are not a general-purpose discussion forum. Please read ... – Okay, I looked at these. It appears that you have a longstanding habit of adding unsourced or poorly sourced claims to Wikipedia and then starting long protest discussions whenever they are reverted. I would urge you to desist from such behavior, which seems like trolling for an argument. It's disruptive to the Wikipedia project, and if you keep at it long enough someone will eventually get annoyed enough to start formal processes demanding that you stop. –jacobolus (t) 13:44, 9 October 2024 (UTC)

Jacobolus, that's quite an unfair conclusion to draw from that discussion! Eric Kvaalen (talk) 06:57, 10 October 2024 (UTC)