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WikiProject Mathematics (Rated B-class, High-priority)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
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High Priority
 Field:  Basics


The template confuses the codomain (which is the set R) with the image (which is [-1,1]). JDiala (talk) 15:36, 14 August 2015 (UTC)


Should this page be combined into Trigonometric Functions as the content is already covered in that page? Wetit🐷 0 03:03, 1 October 2016 (UTC)

The Domain Colouring Picture[edit]

This seems inaccurate. Examining the real line, that is the the x azis, it should move from brightness (absolute magnitude) 0 to 1 and the saturation (complex argument) should be zero, pi, or undefined. I think the hue is actually being used for argument, not saturation. But it still seems wrong.--Mongreilf (talk) 14:18, 28 December 2016 (UTC)


The section currently reads - "Etymologically, the word sine derives from the Sanskrit word for chord, jiva*(jya being its more popular synonym)." - I think this is plainly misleading by conflating proximate and ultimate origins and by confusing the origin of the word with the presumed origin of the concept. Etymology and history need to be clarified. As far as the etymology goes, the immediate word sine is from Latin rather than from the Sanskrit word. I think this section needs to be reworded. Also someone has transcribed the Arabic intermediate but the Sanskrit is not even phonetically transcribed right (it is ज्या in the original - see for example this; jiva seems to be a typo or a misinterpretation - it is usually written as IAST jyā IPA /ɟjɑː/ as given in most reliable sources on Indian trigonometry - example this. In the history section there is a mention of "The chord function" - the link is unhelpful and does not explain what a "chord function" is supposed to mean? It seems that Hipparchus' ideas were based on isosceles triangles - See Carthy, Daniel P. Mc; Byrne, John G. "Al-Khwārizmī's Sine Tables and a Western Table with the Hindu Norm of R = 150". Archive for History of Exact Sciences. 57 (3): 243–266. doi:10.1007/s00407-002-0058-6. ISSN 0003-9519.. Since my own attempts at improvement are reverted, I leave my suggestions here for anyone interested in improving accuracy, referencing and structuring. Shyamal (talk) 18:21, 3 March 2017 (UTC)

There's an inconsistency here with regards to the scholar responsible for the mistranslation anecdote. This page claims it was Gerard of Cremona, however his page makes no mention of the fact, while that of Robert of Chester claims that he did it, with the referance given on the Robert of Chester page quoting two different accounts, one naming Gerard, and the other Robert. Basically it's a mess, and has the smell of an uncheckable legend. I don't have the time to keep digging for an answer, so I'll just throw that out there that all 3 pages need to be brought in line. Fearghalj (talk) 20:53, 24 April 2018 (UTC)

While it may be uncheckable, it is not a legend. There is a priority dispute as to who first translated "jaib" as "sinus". According to D.E. Smith, History of Mathematics (1925), "When Gherardo of Cremona (ca. 1150) made his translation from the Arabic he used sinus for jaib, ..." (Vol. II, p. 616). Some authors (Eves and Maor, in particular) have repeated this and Eves makes it sound like Gherardo originated the translation. However, in a footnote in the same book, Smith says, "The term was probably first used in Robert of Chester's revision of ..." (Vol. I, p. 202), and this is repeated by Boyer. However, Smith is getting his information from A. Braunmühl's Geschichte der Trigonometrie, I (Leipzig, 1900), which discusses this priority issue and brings up the prior use of the term by Plato of Tivoli. Cajori (1906) also mentions Plato of Tivoli as the originator, probably using the same source. It is doubtful that this matter will ever be cleared up and I am not quite sure how we should proceed without giving undue weight to a very trivial dispute. --Bill Cherowitzo (talk) 18:59, 25 April 2018 (UTC)
I have attempted to provide some clarification with a reference. --Bill Cherowitzo (talk) 20:25, 26 April 2018 (UTC)

Derivative in calculus with scalar in radius sine[edit]

Why isn't a scalar included with the calculus chapter? E.g.

For the sine function in radians:

The derivative is:

— Preceding unsigned comment added by JHBonarius (talkcontribs)

Because that is a straightforward application of the chain rule and has nothing to do with the sine function per se. --JBL (talk) 12:59, 1 June 2017 (UTC)

Recent edits[edit]

Laurdecl, to answer the questions in your edit summary: first, the sentence belongs in brackets because it is an aside (this is the article about sine, the other trig functions are incidental). Second, the section on relation to slope has several major issues: its title is wrong (it does not relate sine to slope), none of the key words are defined, it is uncited, and the definition it offers cannot actually be made correct without fully transforming it into the unit circle definition because of issues about which angle and which way to measure the angle (consider the case of a segment with negative slope). I spent some time thinking about resolving these issues, but I don't think that there is any better way than simply removing the section. If you feel there is some important idea in that section not in the unit circle section, I invite you to introduce it there, preferably with a source.

Separately, there is some issue of "angles" versus "measures of angles" -- it is my belief that the input to sine is the measure of an angle, not an angle itself. Do you agree? (This confusion precedes your edits and is present in both yours and mine, but it would be easy to sort out.) --JBL (talk) 12:57, 1 June 2017 (UTC)

I removed the section again because it is very confusing. Normally we think of slope as rise over run, not rise per unit hypotenuse. The concept in the section is not common; hence no citation. I have removed it for now and I do not think it should be reinserted without a reputable citation from a textbook or journal article, indicating that it is in standard use.—Anita5192 (talk) 16:11, 1 June 2017 (UTC)
If you want to format the stuff about related functions as an aside, you could use {{efn}} to do it. It looks bad to have multiple sentences in brackets. I left the slope section in there because I think that it's worth mentioning to the reader as a possible definition of sine. I agree it is poorly worded and doesn't really have anything to do with gradient in a strict sense (unless you substitute run in rise/run with hyp. length). If there really isn't a way to merge it into the section above, then I'm fine with leaving it out, although I think that if it helps one reader then it's worth having. About the angle measure, I see that as purely semantic. Is there a practical difference between 30° and the measure of 30°? Laurdecl talk 09:36, 2 June 2017 (UTC)
I will think about the question of the aside more. 30° is a measure -- it tells you how big something is. By contrast, an angle is a geometric thing determined by some points and lines in the plane; it has a measure, but it is not itself one. --JBL (talk) 12:20, 2 June 2017 (UTC)

Arc length[edit]

Just an extension of the discussion on Talk:Ellipse
The following discussion has been closed. Please do not modify it.

I repost here what was twice vandalized for all editors to see. Both vandals have violated the rules by removing my edits without discussing them on this Talk page.

The arc length of the sine curve between and is This integral is an elliptic integral of the second kind. The arc length for a full period is where is the Gamma function. Alternatively,[1] (See footnote 7 at the bottom of page 1097 of the referenced Notices of the AMS article) it is efiiciently calculated as , where is the arithmetic-geometric mean of and . The reciprocal of is known as Gauss constant. The graph of the sine function might be viewed as a "degenerate" graph of the elliptic function graph. Generally, the length of a period of the graph of an elliptic function is expressed via the modified arithmetic-geometric mean which was apparently introduced for efficiently calculating the length of a thread in a linear parallel force field.[2][3] Cocorrector (talk) 08:21, 12 November 2018 (UTC)


  1. ^ Adlaj, Semjon (September 2012), "An eloquent formula for the perimeter of an ellipse" (PDF), Notices of the AMS, 76 (8): 1094–1099, doi:10.1090/noti879, ISSN 1088-9477
  2. ^ Adlaj, Semjon (2012), "Mechanical interpretation of negative and imaginary tension of a tether in a linear parallel force field", Selected papers of the International Scientific Conference on Mechanics "Sixth Polyakhov Readings", Saint Petersburg, pp. 13–18, ISBN 978-5-91563-110-5
  3. ^ Адлай, Семён (2018-08-30). Равновесие нити в линейном параллельном поле сил. LAP LAMBERT Academic Publishing. ISBN 978-3-659-53542-0.
The contexts, including for the above, are extensively dealt with at Talk:Ellipse. The sophistic claim without discussing does now not hold anymore for this page, too. Purgy (talk) 10:55, 12 November 2018 (UTC)
Four comments:
  1. There's no need to provide an alternative to the expression in terms of the gamma function. This function is among the most well known of the special functions. The alternative expression in terms of the AGM is just a reflection of the special value of .
  2. Reference 1 is a dubious reference. Its main claim, that it provides the first efficient method of computing the complete elliptic function of the second kind, is false. Equivalent methods have been around since the 19th century (see Talk:Ellipse).
  3. The business about the sine function being a degenerate elliptic function is off-topic (references 2 and 3).
  4. Reference 3 is published by LAP Lambert Academic Publishing which is a subsidiary of OmniScriptum; the Wikipedia article describes several problematic practices of this outfit.
cffk (talk) 14:24, 12 November 2018 (UTC)

Ок cffk u seem quite bitter and “a bit we todd did” or is it the other way around? Is that the reason your candidacy will never be considered for any publication at the “AMS Notices”? All your four arguments are dubiously false since not only u're incapable of valuably contributing but u'ant even capable of appreciating valuable contribution of others who stand head and shoulders above you in the math and physics food chain. We all see that had already explained to u the MAGM to a much greater extent than u'll ever care to know, so adding a formula here from that source with “several problematic practices of this outfit” would suffice for your little head. Do not dismiss it since nobody can make it any simpler than this

where this time is a length of a thread in linear parallel force field and is the Jacobi elliptic modulus. With this simple formula and its quite physical interpretation, as I've most recently confirmed from both secondary and primary source, the MAGM was born. Anyone smarter than you from all over the world would easily see that the MAGM appears here without the AGM. Then we all as easily see that you were repeatedly told by the same on Talk:Ellipse that the MAGM can be calculated in two equivalent ways allowing us to appreciate the beautiful formula which was published in An eloquent formula for the perimeter of an ellipse for all of us, including those with mathematical abilities not exceeding mine or even yours, to see. A glimpse at the paper suffices to tell that the author was aware of the equivalence before publishing the paper. Certainly, the equivalence was not discovered by you after the Adlaj publising his paper as you suggest. You comparison is dubious and the Python code you wrote isn't worthy to be included anywhere. U don't even seem to understand yourself since you admitted that Adlaj presented Gauss' method in another way bu you failed to understand the significance of this "other way". What a crippled pitiful soul one would have to attempt hiding or dismissing the formula

and what delusion would lead one to “discover” an error in that awesome beauty and what repeated bout would lead the same one to “double” on that “rediscovery”, in another article, of the same “error” which you u'ant capable of articulating. The formula is beautiful and one has to be quite stupid to argue otherwise. And it is a new 21-st century formula and no one deserves a credit for it aside from its author and Gauss. One has to be quite obnoxious to intervene. Unlike you, the author while fully capable of appreciating the main contribution of Gauss, was capable of appreciating the beauty which u're blind to. Although he did, the author did not need to reference anyone else other than Euler and Gauss. Richard Brent was not as gracious to give Euler and Gauss the credit which they deserved but the truth can’t forever be concealed. The idiot Tom Van Baak thought that Gauss formula of May 30, 1799 was recently discovered as he claimed in his truly dubious article A New and Wonderful Pendulum Period Equation citing Adlaj’s paper as a “Good introduction to elliptic integrals, AGM, and pendulums” without ever reading there that full and all credit was given to Gauss alone. So consistently, u are so oblivious to all this and all the rest of that deviant discussions which are propagated by people as ugly as u. Luckily, there is no permanent place for the ugliness which people like u strive to preserve in mathematics. Your lowest quality followers such as User:Wcherowi and User:Joel B. Lewis are not even able to repeat your arguments since they do not understand them. They just feel and stick to your sick attitude. Most likely they support your pettiness and envy but nothing more. One of them User:Joel B. Lewis has already threatened me with an arranged consensus which seems to unfold here before my eyes. Quite a disgusting environment which can’t permanently last as that beautiful formula would, while exposing lowlifes along its way. So stop wasting your life, the sooner’s the better for u and everyone else. Cocorrector (talk) 19:34, 13 November 2018 (UTC)

@Cocorrector: Your comments are a continuation of the discussion on Talk:Ellipse; please move them there. You should limit the comments on this page to a discussion of Sine. cffk (talk) 20:36, 13 November 2018 (UTC)
Cocorrector, it is deeply shocking that someone as pleasant and charming as yourself is unable to convince other people of the value of your suggestions. Luckily, this mystery is problematic for you alone. --JBL (talk) 22:28, 13 November 2018 (UTC)

Etymology again[edit]

An IP has recently been trying to insert a statement to the effect that the OED claims that there is no Sanskrit origin to the word (or concept if you like) sine. The link provided to justify this leads to a paywall, and I have objected to its use more than once. The IP has failed to understand the intent of my edit summaries and has acted as if I had claimed that the OED was in some sense wrong with this entry. I have just checked the OED and the statement there says that sine comes from the Latin sinus which in turn is how the Arabic jaib (or sometimes written jiba) was translated in the middle ages. The OED makes no claims about the origins of the word jaib. The IP seems to think that this means that this Arabic term did not have Sanskrit origins, but the OED does not say this, so this is just WP:SYNTH. On the other hand, Webster's Unabridged International Dictionary does trace jaib back to the Sanskrit jyā. Together with the fact that all mathematical historians who have weighed in on the subject agree, it strikes me that the IP is just pushing a POV and does not have a real case. --Bill Cherowitzo (talk) 23:03, 14 February 2019 (UTC)

I agree, this is silly, the claim is totally reasonable for inclusion. --JBL (talk) 16:02, 15 February 2019 (UTC)
Bill is completely missing the point. The point of inserting OED reference is to give multiple viewpoints in a scholarly fashion. For the sake of academic diversity of opinions and honesty, one must present different viewpoints rather than make believe a certain authority. There are several stories about the origins of sine and cosine. More detailed version can be found in The Words of Mathematics by Schwartzman. It is quoted here "sine (noun): most immediately from Latin sinus "a curved surface," with subsidiary meanings such as "fold of a toga" and hence the "bosom" beneath the toga; "bay" or "cove." How that word came to represent a trigonometric function is quite a circuitous-and, depending on the authority you believe, contradictory-story. Howard Eves, in his An Introduction to the History of Mathematics, explains the origin of the word in the following way. The Hindu mathematician Aryabhata used the tenn jya, literally "chord," to represent the value of the equivalent of the sine function. When the Arabs translated Indian mathematical works, they transliterated jya as jfba, which actually meant nothing in Arabic. Now Arabic, like Hebrew, is often written with consonants only (pt n th vwls fr yrslf), so jfba became simply jb. Later readers, seeing jb, pronounced it as jaib, which was a real Arabic word meaning "cove, bay." When European mathematicians translated Arabic writings to Latin, they replaced jaib with the Latin word for "cove," which happened to be sinus. The American Heritage Dictionary claims that Arabic jayb (Eves's jaib) did have a meaning, namely "chord of an are," but that Europeans confused the word with the homonym jayb meaning "fold of a gannent," which happened to correspond to Latin sinus. The Oxford Dictionary of English Etymology claims that Arabic jaib meant "bosom," again translated by Latin sinus. For an equally intricate tale of Arabic-Latin translation" Farooq — Preceding unsigned comment added by (talk) 01:15, 16 February 2019 (UTC)
The source you are quoting supports that the word comes from Arabic via Latin, which is what the article says. The source emphatically does not support the claim that "the word sine is not traced to any Sanskrit word" -- it says nothing one way or the other about where the Arabic word comes from. In particular, this is 100% consistent with the (referenced) claim in the article that the chain is Sanskrit -> Arabic -> Latin. --JBL (talk) 01:36, 16 February 2019 (UTC)

Merge proposal: Sine wave into Sine[edit]

I propose to merge the content of Sine wave into Sine as the former article can be adequately expressed within the context of the latter. Jamgoodman (talk) 16:57, 1 July 2019 (UTC)

Deletion of Sine Squared section[edit]

User @Comfr: wants to have a section about the function sin(x)^2; I say this section is redundant as the relationships between sin(x)^2 and the other trigonometric functions are detailed elsewhere, but also the section focuses too specifically on a topic not general enough for the whole article. It would not be appropriate to have a section for sin(2*x), sin(x/2) not sin(x)+1 but they are equally notable as sin(x)^2. I support the redirect and some content in the article referring to sin(x)^2 but an entire section is gratuitous. If Comfr can clarify their position why they think the section should stay, it would be appreciated. Also, please remember that having a redirect does not constitute notability. Plenty of redirects remain on the site despite their articles' deletion. Thanks. Jamgoodman (talk) 08:38, 13 August 2019 (UTC)

I agree that the function does not merit its own section for the reasons listed above and because it is a sinusoid of a different frequency from translated from the origin, which is not significantly different. I think this alone should be mentioned.—Anita5192 (talk) 15:43, 13 August 2019 (UTC)
Sines and squares of sines ofter appear in alternating current equations. As I was attempting to understand an anomalous power factor reading, I began wondering about how squaring affected the shape of a sine function. I assumed that part of a period would be compressed, while another would be stretched. That is all I knew.
I searched for sine squared in Wikipedia, and a redirect took me to the page Trigonometry. Unfortunately, I could not find anything about sine squared on the trigonometry page. I also looked at List of trigonometric identities, Trigonometric functions, and many other searches, without ever finding anything about what a sine squared function might look like.
The breakthrough came when I made a graph of sine and sine squared. I was surprised to see that sine squared was actually a rescaled sine function.
I wish I could have better integrated sine squared into the article, but that was beyond my mathematical ability, so I did the best I could. Perhaps user @Jamgoodman: could identify the specific places where the redundant information already exists, and make those places come up in a Wikipedia search.
Please remember Wikipedia articles should be written so they can be understood by general readers to at least an introductory level WP:TECHNICAL. Many of my colleagues can look at an equation, and immediately see various transformations. I can't.
I thank Jamgoodman for his careful review of my edits. He/she is helping to improve the quality of Wikipedia. Comfr (talk) 03:02, 14 August 2019 (UTC)
I think it would be reasonable to include, in some form (probably not what Comfr originally produced), the double- and half-angle formulas for sine (with an appropriate reference), including the (surprising!) fact that is again a sinusoid. It would be much better to include a proper source, say, a standard textbook on trigonometry or precalculus. One or another version of the formula in question does appear in the (incredibly terrible) article List_of_trigonometric_identities -- check out the section List_of_trigonometric_identities#Half-angle_formulae -- but I don't know how a person who didn't already know what they were looking for would be able to find it. --JBL (talk) 23:16, 14 August 2019 (UTC)
I created the new section sine squared only after I failed to find the information anywhere in Wikipedia. I am not a mathematician, but Wikipedia is a work in progress, so I put what I had discovered into a new section, and expected that eventually more experienced editors would replace it with something better. When I could not find a reference meeting Wikipedia's standards, I supplied this derivation in my original revision. I created the new section to save other readers from the pain I went through. — Preceding unsigned comment added by Comfr (talkcontribs)
Yes, you already made this clear (though the information is already in WP, as I noted). --JBL (talk) 11:16, 15 August 2019 (UTC)