Talk:Sine and cosine

Thoughts on renaming article "sine and cosine"

I'd like to preface by saying this is the only trig function with its own dedicated article.

It's been suggested that this article be merged with the Trig functions article. While I wouldn't be opposed to this, I have an alternative suggestion which I think should at least be considered.

I'd like to mention that sine and cosine are by far the most widely used trig functions. They're used for converting from polar, finding parallel & perpendicular components, rotating reference frames, Fourier transforms, splitting exponents into real and imaginary parts, solving a wide variety of differential equations. These are just off the top of my head, and I could go on, really.

I initially thought, "so, why don't we give cosine it's own article as well?" But, I realized sine and cosine are extremely similar. Their graphs are shaped the same, they do similar things (albeit in opposite directions), and they're often used interchangeably (i.e. y=sin(x+π/8) is the same as y=cos(x-3π/8)). Of course, if cosine had its own article, I wouldn't be opposed to that either.

There are plenty of viable solutions to this, but I think many of us can agree, it seems kinda silly to just have one article on sine. But at the same time, there's a lot of info here exclusive to the sine, and merging with another article might be a lot of work.

I personally would be very much on board with changing this article to "sine and cosine", and inserting some extra info about cosine, but I'm open to other ideas as well. Math Machine 4 (talk) 21:00, 26 October 2020 (UTC)

This strikes me as not a bad idea. --JBL (talk) 21:51, 26 October 2020 (UTC)

@Math Machine 4: Totally agree. I've created a draft at Draft:Sine and cosine, but I've only added cosine to the lead and infobox so far. Danstronger (talk) 03:51, 13 November 2021 (UTC)

:) Glad I could make a difference Math Machine 4 (talk) 20:45, 12 July 2022 (UTC)

Bug?

In the "Arc length" section, for some reason, "from $0$ to $x$" is displayed as "from to $x$". It is quite confusing. Is it just me, or is someone else experiencing this issue? A1E6 (talk) 13:51, 22 August 2021 (UTC)

Yep. It's the math-0 bug. See Wikipedia:Village_pump_(technical)#Math_0. I added a space to solve it. - DVdm (talk) 14:30, 22 August 2021 (UTC)

Propose moving page to "Sine and cosine"

As mentioned above, since there is no page for cosine, I think this page should be moved to "Sine and cosine" and information about cosine should be integrated into it. I have created a Draft:Sine and cosine as a proposed new version of the page. Danstronger (talk) 03:22, 17 November 2021 (UTC)

Infobox

It doesn't make sense to call the Gupta period the "date of solution" of the sine and cosine functions, since they are functions, not problems or equations. That being said it's difficult to change because of the mechanics of this infobox. It also seems like the box is a collection of random facts that really doesn't convey important information, except maybe if you're a student trying to look for homework answers. I think it should be removed or made much smaller, and the most important information simply stated in the lead. Wuffuwwuf (talk) 15:33, 14 February 2022 (UTC)

• Ok. I'm proposing to get rid of everything in the infobox after and including the "Domain and Range" section. Wuffuwwuf (talk) 14:26, 15 February 2022 (UTC)

Richard Feynman's notation

Might be worth adding a note about Richard Feynman's trigonometry notation? See example: https://twitter.com/fermatslibrary/status/1512788437199462409 —Pengo 06:56, 10 April 2022 (UTC)

Adlaj material

Per extensive discussion on Talk:Ellipse/Archive 2 and this from my talk page, I do not believe the Adlaj paper should be used here. Also, @A1E6:, note that per WP:ONUS The onus to achieve consensus for inclusion is on those seeking to include disputed content - this content should be left out until consensus is gained for inclusion. MrOllie (talk) 23:31, 18 June 2022 (UTC)

The version you undid is very different from the 2016 version. Please do not blindly refer to lengthy old discussions; state precisely what Wikipedia policies is the content going against and why.

Let me summarize:

1) The article was removed because of self-promotion; when I added it in 2021 I had no knowledge of this Adlaj drama that happened 5 years before; I'm not an Adlaj sock

2) The article is from Notices of the American Mathematical Society, a reliable source

3) The article is cited by at least two other papers [1][2].

4) I think the content improves the Sine and cosine article because the convergence is quadratic (very fast). From a paper [3] by Borwein and Zucker "The reverse procedure of expressing the gamma function in terms of complete elliptic integrals allows us to use the arithmetic-geometric iteration to compute the gamma function, using quadratically convergent iterations." A1E6 (talk) 23:38, 18 June 2022 (UTC)

Since no additional support has showed up, I have re-removed this per WP:ONUS, there is no consensus for inclusion. MrOllie (talk) 21:09, 20 June 2022 (UTC)

@MrOllie: I will ask again: what Wikipedia policies is the content going against and why? And who determined that the content doesn't improve the article? A1E6 (talk) 22:52, 20 June 2022 (UTC)

We discussed this on my user talk page, you can re-read my concerns there. MrOllie (talk) 23:06, 20 June 2022 (UTC)

@MrOllie: I refuted every problem you were pointing out. You decided to blindly repeat your concerns instead of replying to the refusals – this way of argumenting is immature at best. A1E6 (talk) 23:49, 20 June 2022 (UTC)

That's what happens when your responses don't convince the other party they are incorrect. MrOllie (talk) 00:03, 21 June 2022 (UTC)

@MrOllie: Why did it not convince you, then? A1E6 (talk) 01:04, 21 June 2022 (UTC)

Because you weren't very convincing - and my objections that this is an obscure result that isn't worth covering still stand. FYI, the fact that I'm not responding immediately or that I prefer to refer to the discussion we had on another page rather than replaying it here does not mean I'm 'unwilling to communicate' - more like unwilling to repeat myself multiple times in a day. MrOllie (talk) 20:28, 21 June 2022 (UTC)

You purposely don't address new points other editors make and instead you keep repeating the same stuff, over and over again. A1E6 (talk) 20:44, 21 June 2022 (UTC)

I'm not seeing new points here, just repetitions of the same stuff (unless we count personal attacks as new points). MrOllie (talk) 20:47, 21 June 2022 (UTC)

The article is from a reliable source and it is cited by at least two other papers, so it is not "obscure" and I think it is worth covering. A1E6 (talk) 20:50, 21 June 2022 (UTC)

As I see it there are two quotes from Adlaj in the disputed material. First is the arithmetic-geometric mean calculation for the arc length of a full period. IMO this is not very useful - the arc length is a constant. Practically I imagine most people will simply copy what's there instead of calculating it. The article only gives float precision though, I changed it to 7.640395578055424 for full double precision (just plugged it into Wolfram Alpha). Anyways, if someone does need to calculate it, the gamma function is going to be available in an arbitrary precision library, and suffices. Anything more efficient is complete overkill for an article whose main subject is sines and cosines. And the fast algorithms are cited in the linked articles:

• The Borwein and Zucker paper is cited in Gamma function#Approximations
• Elliptic integral#Complete elliptic integral of the first kind cites Carlson for arithmetic-geometric mean which is according to MrOllie is better than the formulas by Adlaj. If Adlaj's formula was going to be added it would be here IMO.

The other quote is that L is the circumference of a certain ellipse. The meaning of "incomplete elliptic integral" is not obvious, whereas circumferences are taught in grade school. In fact the section Ellipse#Circumference and the following one on arc length discuss the elliptic integrals somewhat. As the elliptic integral article does not have this information at all, I think this link and its associated information is useful. --Mathnerd314159 (talk) 01:17, 21 June 2022 (UTC)

@Mathnerd314159: "IMO this is not very useful - the arc length is a constant." – You've got to be kidding. For example, there is a whole article dedicated to Approximations of π (a constant!). The Carlson formula is for computing the elliptic integral of the second kind for general arguments and in some cases it can be simplified (the resulting formula is the one that MrOllie removed), please see Elliptic integral#Computation. It seems that you accidentally mixed up the complete elliptic integral of the first kind and the complete elliptic integral of the second kind in your reply. A1E6 (talk) 01:41, 21 June 2022 (UTC)

The corresponding article for π would rather be List of formulae involving π, as nobody has bothered to compute L to trillions of digits in a competition to burn CPU cycles. But when I google the value of L the two descriptions I get are "the arc length of sin between 0 and 2*pi" and "the circumference of an ellipse with semi-axes 1 and sqrt(2)", and that's it. There are only 2 formulae involving L, and they're both mentioned in this article. So no need for a new article List of formulae involving L. (Not to mention that L is a terrible name- AFAICT this constant doesn't even have a proper name)

It does seem that the elliptic integral is of the second kind, they both cite Carlson so I wasn't really picky on the link. Mathnerd314159 (talk) 03:16, 21 June 2022 (UTC)

No one called for List of formulae involving L. I used a letter for the constant so that we can refer to it easily throughout the text, I think that was reasonable, there's no better way of doing that. A1E6 (talk) 08:42, 21 June 2022 (UTC)

Well, you picked right, L is the de-facto name. It is used in in this SE answer, this website, in this SO question, this mailing list thread, this book, these course notes, and this book (as L(W)). For comparison the non-L usages I found were I, S, s, s and lenght. With the book sources and including the textbooks with sin from 0 to pi there is probably enough material for notability of this constant. But I don't know what the article would be called. Mathnerd314159 (talk) 23:50, 21 June 2022 (UTC) edited 04:19, 25 June 2022 (UTC)

• I support MrOllie's version: The section contains three formulas, none is proved nor computed in the article. So, we do not need a special treatment for the second formula alone. The way of computing the numerical values of these formulas is is out of scope for several reasons. Firstly, they are too technical. Secondly, they belong to articles on the involved special functions. Finally there are no reason to favour a specific method of computation. As far as I know, the arc length of the sine is a holonomic function, and there are efficient softwares for computing automatically the value of any holonomic function to any desired accuracy. I do not know whether Adlaj method is faster or not than the direct computation with holonomic methods, but the comparison does not belong to this article. D.Lazard (talk) 15:24, 22 June 2022 (UTC)

  There's no comparison: in my version, the method is described as "very rapid" (it really is, the arithmetic–geometric mean iterations are known to be quadratically convergent) and is not compared to anything else, it is not "favoured".

  Why do you think that the way of computing is too technical? It is a section dedicated to the arc length of the sine curve.

  Regarding "proofs" – The proof is so trivial that the reader is supposed to observe it. Also let me quote Borwein and Borwein: "The evaluation of $E(1/{\sqrt {2}})$ is left as Exercise 1" (Pi and the AGM, p. 25). A1E6 (talk) 15:34, 22 June 2022 (UTC)

• I don't see a need to include the Adlaj paper here; it's a curiosity about a niche-interest subtopic (arc length) of a topic with wide readership. XOR'easter (talk) 15:53, 22 June 2022 (UTC)

  How does the fact that it is a niche-interest subtopic justify the first part of your statement? A1E6 (talk) 16:09, 22 June 2022 (UTC)

  An encyclopedia requires a neutral point of view, and WP:Neutral point of view is a fundamental policy of Wikipedia. This implies that a niche-interest subtopic must receive its WP:due weight. This is the clear justification of the whole XOReaster's statement.

  Also, talking of a method without mentioning the existence of other methods is favouring your preferred method, or, if you prefer the standard Wikipedia terminology, giving it a undue weight. Again, this is against the policy WP:NPOV. D.Lazard (talk) 17:09, 22 June 2022 (UTC)

  It comes from a reliable source and there are at least two other papers citing the article, so why do you think it doesn't receive its due weight?

  I'm not talking about "my" method without mentioning the existence of other methods, don't you see the gamma function method (right above it)? A1E6 (talk) 17:32, 22 June 2022 (UTC)

  As mentioned in our discussion on my talk page, Wikipedia is not an indiscriminate collection of everything that has appeared in a reliable source. Almost every paper gets some small number of citations somewhere, that is not a reason that it must be included on Wikipedia. MrOllie (talk) 17:38, 22 June 2022 (UTC)

  I think it is relevant and that it improves the arc length section. You changed your reason for removal several times (this doesn't seem very convincing if you ask me):

  First: a reference to an old self-promotion discussion (I'm not an Adlaj sock and had no knowledge of it when I added the content + my version is very different from the 2016 version) and stating that "the 'very rapidly' is unjustified" (it is not unjustified, as the convergence of the arithmetic–geometric mean iterations is quadratic (very fast)).

  Second: "no one writes about it" (which is not true, it is cited by at least two other papers)

  Third: "WP:UNDUE is commonly used to exclude things that aren't commonly referred to by others" (not true, this stuff is not even in WP:UNDUE)

  Fourth: "Since no additional support has showed up, I have re-removed this per WP:ONUS" (actually, additional support for the article showed up afterwards, Mathnerd314159)

  When I asked you why did my arguments not convince you, you simply stated "Because you weren't very convincing" – this has to be a joke response. A1E6 (talk) 18:41, 22 June 2022 (UTC)

  My reason for removal has been WP:UNDUE the whole time - this is a niche detail that does not merit mention in the article. If my phrasing has varied it is because the questions being asked have varied, and I thought that cutting and pasting the same response would have been a little rude. Also, I do not believe that these out of context quotes are a fair reflection of the conversation. - MrOllie (talk) 18:44, 22 June 2022 (UTC)

  The "Arc length" section is a niche subtopic and I expanded the section a little bit, what more than a reliable source and two citations do you expect? A1E6 (talk) 18:49, 22 June 2022 (UTC)

  Yes, the arc length of the graph of the sine function is a niche topic which probably doesn’t need to be included at all. But I don’t feel too strongly about it. –jacobolus (t) 19:30, 22 June 2022 (UTC)

  Mathnerd314159 found at least three books dealing with the arc length of the sine curve, so I disagree. A1E6 (talk) 19:33, 22 June 2022 (UTC)

I think MrOllie’s version is still too much. I would propose cutting to just:

Arc length of the graph of the sine function

The arc length of the graph of the sine function from ${\textstyle 0}$ to $x$ is

$$\ell (x)=\int _{0}^{x}\!{\sqrt {1+\cos ^{2}(u)}}\,du={\sqrt {2}}E{\Bigl (}x,{\tfrac {1}{\sqrt {2}}}{\Bigr )},$$

the arc length from ${\textstyle x_{0}}$ to ${\textstyle x_{1}}$ is ${\textstyle \ell (x_{1})-\ell (x_{0})}$, and the arc length for a full period is

$$\ell (2\pi )={\frac {4{\sqrt {2\pi ^{3}}}}{\Gamma {\bigl (}{\tfrac {1}{4}}{\bigr )}^{2}}}+{\frac {\Gamma {\bigl (}{\tfrac {1}{4}}{\bigr )}^{2}}{\sqrt {2\pi }}}=7.640395578\ldots ,$$

where $E(\varphi ,k)$ is the incomplete elliptic integral of the second kind with modulus $k$ and $\Gamma$ is the gamma function.

And the section should include a figure so that readers know what is being described (a simple figure can be made with Desmos). Any further details about calculation should go to the page about incomplete elliptic integrals of the second kind, where they are more relevant/appropriate (or possibly into a footnote). –jacobolus (t) 19:58, 22 June 2022 (UTC)

There's no better place for a rapid computation method of the arc length constant other than the arc length section. The arc length section is very short and a fast computation method of the constant is relevant, so why do you suggest footnotes? A1E6 (talk) 20:18, 22 June 2022 (UTC)

If it doesn’t belong there, and there’s no better place for it, maybe it would be fine to leave in a paper at arxiv or something. The part of this section about a periodic correction to ${\textstyle Lx/2\pi }$ is also super overkill here. In practice if someone wants to compute this function rapidly (and doesn’t want to just call the most easily available incomplete elliptic integral implementation), they are going to do domain reduction to ${\textstyle [0,{\tfrac {1}{2}}\pi ]}$ and then evaluate on that domain using a polynomial or rational approximation with coefficients chosen to minimize maximum error across the interval, of high enough degree to meet their acceptable accuracy tolerance. No more than a handful of people in the world have ever needed an efficient approximation to the arclength of the graph of the sine function, and all of them are capable of doing a search of the academic literature; this is not of broad enough interest to demand space and attention in an article like sine and cosine aimed mostly at a lay audience. –jacobolus (t) 20:47, 22 June 2022 (UTC)

"If it doesn't belong there [...]" – I think it belongs to the arc length section, I already gave reasons for it; you think it's "super overkill"...

Regarding the periodic correction (Fourier series) – the convergence is quite fast; one can even use that series together with the domain reduction to $[0,\pi /2]$, there's no need for rational approximations chosen to minimize maximum error; but let me remark that this section is intended for discussing Adlaj material and the Fourier series is not Adlaj material. A1E6 (talk) 21:16, 22 June 2022 (UTC)

The performance of off-the-shelf implementations of the incomplete elliptic integral are also perfectly sufficient for most purposes (and the 4 people in the world who ever need something fancier are surely capable of finding this information outside of Wikipedia or figuring something out for themselves). If someone needs something faster they aren’t going to reach for a trigonometric series; the polynomial approximation is going to be finished before you get through the first trig function evaluation (a degree 26 polynomial approximation of this function on ${\bigl [}0,{\tfrac {1}{2}}\pi {\bigr ]}$ is accurate to machine precision for double-precision IEEE floats; evaluating a degree 26 polynomial is incredibly cheap). And someone trying to evaluate “L” is going to just declare it in their code as a constant. –jacobolus (t) 21:26, 22 June 2022 (UTC)

Please try to focus on the dispute between me and MrOllie (Adlaj material), not the Fourier series. A1E6 (talk) 21:31, 22 June 2022 (UTC)

The goal should be to leave the article in an effective consensus state following Wikipedia guidelines, not just make a narrow resolution that results in a subsequent conflict immediately afterward. For me, reasonable options include (a) cutting this section very short, basically just pointing people toward elliptic integrals and optionally including a few more details and some references in a footnote (I have no problem with listing Adlaj as one of those references), or else (b) removing the arclength section entirely, or relocating it to sine wave. –jacobolus (t) 21:41, 22 June 2022 (UTC)

Aside: I implemented a polynomial approximation accurate to machine precision. I can evaluate 12.5 million arclengths per second in my web browser (a properly vectorized native-code implementation of Clenshaw's rule would get at least 10x speedup from there). Finding the coefficients took 1 line of code using Chebfun. Link: https://observablehq.com/@jrus/sine-graph-arclength –jacobolus (t) 22:15, 22 June 2022 (UTC)

(edit conflict) This page is not the place for a dispute between two editors. This thread in particular is about the content of the section on the arc length. It is not reasonable to split the discussion in as many threads as there are paragraphs in the section. In any case, there is a clear consensus against your edits, and if there is a dispute, it is between you and all other editors of this thread. Please take this into consideration. D.Lazard (talk) 21:53, 22 June 2022 (UTC)

This section is about Adlaj material, not the arc length section as a whole. It's not true that a dispute is between me and all other editors: Mathnerd314159 supported the inclusion of the article. A1E6 (talk) 22:08, 22 June 2022 (UTC)

According to Mathnerd314159,

"As the elliptic integral article does not have this information at all, I think this link and its associated information is useful."

Should this

"$L$ is the circumference of an ellipse when the length of the semi-major axis equals ${\sqrt {2}}$ and the length of the semi-minor axis equals $1$."

be included? Does anyone else have an opinion on this? A1E6 (talk) 19:58, 24 June 2022 (UTC)

Which is the “this information” you are thinking about, specifically? I don’t think it needs to elaborate about the ${\sqrt {2}}$ case specifically, but the elliptic integral article could certainly be massively improved. Mentioning in this arc length section of sine and cosine that “the arc length for a full period is equal to the circumference of the ellipse ${\tfrac {1}{2}}x^{2}+y^{2}=1$” seems fine to me. One sentence is enough for this; any more technical details can go in a footnote. –jacobolus (t) 23:12, 24 June 2022 (UTC)

Information about the arclength of the graph of the function would probably be more appropriate at sine wave (a.k.a. sinusoid), but generalized to handle a sinusoid with any amplitude ${\textstyle (x,a\sin x)}$. –jacobolus (t) 21:02, 22 June 2022 (UTC)

To be verifiable this would require source(s) calculating the arc length of a sinusoid, but I googled and didn't find anything usable. The closest was [4] which does an example but doesn't mention elliptic integrals. The arc length for $A\sin(\omega T)$ is ${\frac {\sqrt {a^{2}\omega ^{2}+1}}{w}}E\left(t\omega ,1-{\frac {1}{a^{2}\omega ^{2}+1}}\right)$ which is quite long compared to the one for sin, so it seems unlikely someone published a reliable source with this formula. Mathnerd314159 (talk) 04:23, 25 June 2022 (UTC)

• My vote: jacobolus's version with the "circumference of the ellipse ${\tfrac {1}{2}}x^{2}+y^{2}=1$" note added. --Mathnerd314159 (talk) 05:23, 25 June 2022 (UTC)

  Actually, new plan: remove the gamma equation entirely. Just link to the section I added: Gauss's constant § Circumference of an ellipse. Something like "The arc length for a half period is 3.820197789..., a value related to Gauss's constant." The gamma equation in this article follows from the relations $\Gamma {\bigl (}{\tfrac {1}{4}}{\bigr )}={\sqrt {2G{\sqrt {2\pi ^{3}}}}}$ and $C={\frac {1}{G}}+G\pi$ given in the Gauss constant article and Gauss's constant makes the equations simpler. Mathnerd314159 (talk) 07:11, 25 June 2022 (UTC)

  So I propose this:

  Just add "$L$ is also related to Gauss's constant". I certainly wouldn't remove the gamma expression, as the gamma function is known much better than Gauss's constant.

  Also @Mathnerd314159: When you added Adlaj's article to Gauss's constant, you cited a certain sentence – is that really necessary? I think it is just confusing for a new reader (when it's taken out of context), as Adlaj uses $L$ for the lemniscate constant (a different thing). A1E6 (talk) 10:52, 25 June 2022 (UTC)

  The page is quite dense, so the quote is necessary to show that we are citing the footnote. It is a little confusing that Adlaj uses different notation, but most readers don't read the footnotes at all so I think the location benefits of the quote outweigh any confusion. Mathnerd314159 (talk) 15:16, 25 June 2022 (UTC)

Software implementations (why 2008)?

The "Software implementations" section makes a specific reference to the 2008 revision of IEEE 754 and links to a special (limited) page for it (https://en.wikipedia.org/wiki/IEEE_754-2008_revision). Is there any particular reason to refer to the 2008 revisions (there is also a 2019 revision), rather than to just link to the general 754 standard, which has more substance?

Namely --> https://en.wikipedia.org/wiki/IEEE_754#Recommended_operations DKEdwards (talk) 19:21, 20 November 2022 (UTC)

I have fixed the link. I have also clarified the first paragraph: IEEE 754 does not contain any implementation; this is a standard for specifying the input output behavior of implementations. D.Lazard (talk) 21:09, 20 November 2022 (UTC)

Expand 'Pythagorean trigonometric identity'

There isn't much about 'sin^2+cos^2=1' on this page. We can expand this topic. Bera678 (talk) 16:31, 21 December 2023 (UTC)

Yes, there is: Sine and cosine#Pythagorean trigonometric identity. I know of no reason to expand it.—Anita5192 (talk) 15:38, 24 December 2023 (UTC)

What more information do you think is worth adding Bera678? JBW (talk) 21:37, 2 January 2024 (UTC)

Etymology

The original text was

Etymologically, the word sine originated in the Sanskrit word jyā

This is a linguistic nonsense. Etymologically sine originlated in sinus. However if you want it a historicl sequence, then one may write

he word sine originated in the translation of the Sanskrit word jyā

or something like that. - Altenmann >talk 20:43, 12 May 2024 (UTC)

This seems like an ideosyncratically extremely narrow definition of "etymology". A dictionary definition, "the study of the origin of words and the way in which their meanings have changed throughout history", clearly encompasses this topic. –jacobolus (t) 20:52, 12 May 2024 (UTC)

I see no relevance in your remark. I didn't rename the section, do I? - Altenmann >talk 00:58, 13 May 2024 (UTC)

I feel like your comments are very aggressive/hostile ("linguistic nonsense", etc.), and I don't really understand why. This story is not nonsense. It's a helpful basic explanation of how this English word came to its current meaning. –jacobolus (t) 01:14, 13 May 2024 (UTC)

There's a decent account with several sources in https://jeff560.tripod.com/s.html, search down for "sine". –jacobolus (t) 21:04, 12 May 2024 (UTC)

Tripod.com is not a source of expertise. - Altenmann >talk 00:58, 13 May 2024 (UTC)

To the contrary, this is one of the best available sources anywhere about the history of mathematical names. While mathematics teacher Jeff Miller is only an "amateur" historian, tracing down the origins of mathematical words is a pretty niche topic and frankly I'm not sure there are any living scholars specializing in it. His site is quite comprehensive and carefully researched. If you spend any time at all studying this topic, you will come across it repeatedly. –jacobolus (t) 01:12, 13 May 2024 (UTC)

Opening the black box in the lede

@Anita5192: thanks for your edit comment. I don’t understand it though. “Algebra” and “Taylor series” are not mutually exclusive terms - algebra is nothing more than the replacement of a number with a variable.

Please could you explain your aversion to showing readers what is inside the black box in the lede? I cannot think of a more impactful way to demystify the topic than by showing the simple expansion of sin and cos in its most basic form.

Onceinawhile (talk) 08:01, 4 June 2024 (UTC)

Many important things can be said about sin and cos. Not all them can go in the lead. Whether "described algebraically" is correct is not the point and my view would be that the expansion is not needed there. Johnuniq (talk) 08:39, 4 June 2024 (UTC)

I agree with Johnuniq comment that exapnds the second point of Anita's edit summary. About the first point: in common terminology, infinite operations, such as series are not considered to belong to algebra, but they belong to analysis. In paricular, sine and cosine are not algebraic functions since they cannot be defined algebraically. D.Lazard (talk) 09:32, 4 June 2024 (UTC)