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== Algebraic value of sin 45° ==

{{ping|DVdm‬}} reverted my change from

:<math>\sin \frac\pi4 = \sin 45^\circ = \frac{\sqrt{2}}{2} = \frac{\sqrt2}2</math>

to

:<math>\sin \frac\pi4 = \sin 45^\circ = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt2}</math>

with the comment "Sqrt(2)/2 is much more common than 1/sqrt(2) in the literature". Though I understand that &radic;2/2 is more common, the line gives it twice. I wonder if it may help learners to know that both expressions are valid, should they come across the rarer form. Does anyone have any thoughts on this?

Cheers,<br />'''[[User:cmglee|cm&#610;&#671;ee]]'''&#9094;[[User_Talk:cmglee|&#964;a&#671;&#954;]] 22:55, 5 July 2020 (UTC)
:{{u|Crissov}} added the accidentally duplicate values on [[Special:Diff/937238790|23 January 2020]]. Previously, the values looked [[Special:PermanentLink/937232343#Algebraic values|like this]] and the duplication probably came from the "easy way to remember" values. Probably best to omit. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 23:57, 5 July 2020 (UTC)
::Yes, I already removed the duplicate values: [https://en.wikipedia.org/w/index.php?title=Trigonometric_functions&type=revision&diff=966142213&oldid=966141888]. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 06:33, 6 July 2020 (UTC)
:::It might make sense to add a note to show it either way [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 12:42, 17 December 2023 (UTC)

== Notes and References ==

In the article, there are references in the notes section and another section of references in the references section. This error (if it is an error) makes it impossible to annotate the article. Please someone fix this bug [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 14:01, 17 December 2023 (UTC)

:There are a variety of reference styles used in Wikipedia articles, and this is one of the more common ones. If you add a source which will be used repeatedly, especially a long source with separate page numbers for the different claims, put it into 'references' and then cite it with a shortened citation in the footnotes. If you add a source used for just one claim, or a source used a few times but which is short enough to not need a page specified or where the pages used are the same for all cited claims, put it directly into a footnote. Feel free to also add textual notes to the 'notes' section. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 16:09, 17 December 2023 (UTC)
::But this prevents adding notes to the article? [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 16:48, 17 December 2023 (UTC)
:::I don't understand what you mean. What kind of note are you trying to add? Feel free to mix textual notes in with the reference footnotes. If you get consensus here, e.g. if you plan to do a substantial rewrite of the article, you can probably do some amount of reformatting of the appendices. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 16:51, 17 December 2023 (UTC)
::::To add notes to the article, the <nowiki>{{notelist}}</nowiki> command must be used in the notes section. [[User:Bera678|Bera678]] ([[User talk:Bera678|talk]]) 08:47, 19 December 2023 (UTC)
:::::Feel free to add textual notes to the numbered footnotes currently in the article. You can add these with <code><nowiki><ref>...</ref></nowiki></code>. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 09:11, 19 December 2023 (UTC)
== "[[:Logarithmic sine]]" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] ==
[[File:Information.svg|30px]]
The redirect <span class="plainlinks">[//en.wikipedia.org/w/index.php?title=Logarithmic_sine&redirect=no Logarithmic sine]</span> has been listed at [[Wikipedia:Redirects for discussion|redirects for discussion]] to determine whether its use and function meets the [[Wikipedia:Redirect|redirect guidelines]]. Readers of this page are welcome to comment on this redirect at '''{{slink|Wikipedia:Redirects for discussion/Log/2024 January 29#Logarithmic sine}}''' until a consensus is reached. <!-- Template:RFDNote --> [[User:1234qwer1234qwer4|1234qwer]][[User talk:1234qwer1234qwer4|1234qwer]][[Special:Contribs/1234qwer1234qwer4|4]] 23:22, 29 January 2024 (UTC)

== tg should not be used. ==

tg and ctg, arctg and arcctg should not be used in accordance with ISO IEC 80000-2:2009. That is absent at the article. [[User:Voproshatel|Voproshatel]] ([[User talk:Voproshatel|talk]]) 19:01, 4 February 2024 (UTC)

:yeah i also kind of agree [[User:Hamterous1]] ([[User talk:Hamterous1|discuss anything!🐹✈️]]) 00:25, 17 April 2024 (UTC)
:Different countries and different authors had/have different conventions about this. In modern English language sources "tan" is the most common, partly because it is the only one supported by default in LaTeX. But if you look at work from France, Germany, or Russia, especially historical sources, you will commonly find "tang" and "tg" as an alternative. The current text in the article is fine, but if someone can find a clear discussion of this it would also be fine to more explicitly describe the relative popularity and extent of these various symbols. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 01:43, 17 April 2024 (UTC)
::Oh, ok [[User:Hamterous1]] ([[User talk:Hamterous1|discuss anything!🐹✈️]]) 11:50, 17 April 2024 (UTC)

== Offensive content ==

I am astounded that Wikipedia allows the mnemonic "All Science Teachers (are) Crazy". The term "crazy" isn't acceptable, is it? Sure it's outrageousness makes it memorable, but I can think of lots of unacceptable mnemonics. Like the resistor code Bad Boys Ravished (raped) Our Young Girls But Violet Gave Willingly was taught to me in college, believe it or not. (Black, Brown, Red, Orange, Yellow, Green, Blue, Violet, Gray, White - I still remember it 60 years later...) Anyway, I was taught, even earlier than the resistor color code, "All Students Take Calculus" which perhaps isn't as useful today, since trig is (apparently) no longer the "gateway" into precalculus. I suggest an encyclopedia isn't an appropriate place for insults, even if the intent isn't malicious.[[Special:Contributions/71.31.145.237|71.31.145.237]] ([[User talk:71.31.145.237|talk]]) 11:29, 21 August 2024 (UTC)

:Wikipedia does not insult by saying that all science teachers are crazy. It reports (with a reliable source) that some people use that [[Mnemonics in trigonometry|mnemonic in trigonometry]]. Just like Wikipedia does not say that Trump is a liar. It gives a list of reliably documented [[false or misleading statements by Donald Trump]]. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 11:52, 21 August 2024 (UTC)
:Perhaps we could replace "All Science Teachers (are) Crazy" with "All Students Take Calculus".  The latter is slightly better to me as a mnemonic because it doesn't have the extra word "are", and it has the added advantage of avoiding the potentially offensive language.  Personally, my general rule is to say what needs to be said even if it is offensive, but if you can say what you need to say without being offensive then please do!  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 14:21, 21 August 2024 (UTC)
:See [[WP:NOTCENSORED]]. But in any event, this is awfully mild and lighthearted to be called "outrageous", given all of the actually outrageous things in the world. (By comparison, your mnemonic joking about rape is quite gratuitously crass and sexist, and perhaps racist, and any teacher presenting it today could expect to be fired.) –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 16:02, 21 August 2024 (UTC)
:Personally, I think "All Science Teachers (are) Crazy" is pretty tame as far as old mnemonics go; even if "crazy" is one of those stigmatizing words we should be making an effort to say less often, it's not what I'd call ''outrageous.'' But Quantling makes a good point that "All Students Take Calculus" is better on other grounds. Moreover, poking into the literature, it doesn't appear that people actually favor one mnemonic over another to a great extent here, so we shouldn't imply that only one mnemonic exists. I've gone ahead and swapped it out, bringing over the reference from [[Mnemonics in trigonometry]]. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 23:04, 21 August 2024 (UTC)
::Interesting. I didn’t realise multiple mnemonics had been created. The one I learned as a 16-year old was “All Stations To Claremont” where Claremont was a well-known station on the railway line that served the largest city in the vicinity. [[User:Dolphin51|<i style="color: green;">''Dolphin''</i>]] ''([[User talk:Dolphin51|<span style="color: blue;">t</span>]])'' 03:15, 22 August 2024 (UTC)

== Bartle and Sherbert ==

The bibliographic details for [[Robert G. Bartle|Bartle]] and Sherbert are missing. I believe their ''Introduction to Real Analysis'' went through something like four editions; which one was used here? [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 19:48, 23 August 2024 (UTC)

: 3rd edition, page 247. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 19:58, 23 August 2024 (UTC)
::OK, added. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 21:04, 23 August 2024 (UTC)
P.S., does anyone have access to 
* Landau, Edmund (1934). Einführung in die Differentialrechnung und Integralrechnung (in German). Noordoff.  
It seems to me that the approach we are crediting to Bartle and Sherbert is very likely due to Landau.  It would be nice to include a reference if so.  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 14:05, 25 August 2024 (UTC)

== "Definition via integration" section ==

[[File:Arc, chord, half-tangent, versine, sine.png|thumb|The half-tangent among other "trigonometric lines"]]

@[[User:Tito Omburo|Tito Omburo]] I'm still a bit concerned about this new section. I think it's a bit misplaced and somewhat misdirected. The half-tangent is well worth mentioning somewhere here, but the "analysis" section is not the right place in my opinion, and I think the way this one is currently written gives this undue weight in this context. I think any such discussion should clearly and explicitly point out that (a) the half-tangent is itself a "trigonometric function" of angle, and (b) it's an alternative representation to angle measure, in terms of which other trigonometric functions can be described rationally, making it to a substantial extent a way to ''avoid'' trigonometric functions and calculus/analysis, in favor of algebra.

(Aside: I got quite carried away with a draft [[User:Jacobolus/HalfTan]], which grew far beyond reasonable article scope and should probably be published somewhere external to Wikipedia. It will take a lot of work, especially trimming, to salvage some parts as a Wikipedia article at this point, and I got quite stalled on the project.)

But in any case, if we're going to add a section about this, its details should be well sourced and its content should be broadly reflective of the way the topic is addressed in high-level sources about trigonometric functions or trigonometry. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 17:50, 24 August 2024 (UTC)

: I agree the analysis section should be later in the article.  Hardy explicitly defines the trigonometric functions by integration, although he uses the tangent substitution which is algebraic rather than rational.  Bourbaki defines the trigonometric functions using unitary representations of the torus, as I also recently added.  These seem like reasonable high-level sources for the article.  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 18:00, 24 August 2024 (UTC)
::I think the section as currently stands violates the spirit of both [[Wikipedia:No original research]] and {{slink|Wikipedia:Neutral point of view|Due and undue weight}}, as a section written from scratch by a Wikipedian and only loosely related to a couple of historical sources and not discussed in this manner in common survey sources about the topic. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 18:11, 24 August 2024 (UTC)
:::I don't think it's undue weight to include several definitions of the trigonometric functions that have been used in analysis.  I used Hardy, in particular, because he addresses the unsatisfactory nature of the usual definition in elementary calculus, and someone had complained about it.  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 18:42, 24 August 2024 (UTC)
::::The development you are presenting is not Hardy's though, but your own, for which you don't have any source. At best this is "original synthesis".
::::It's not bad as exposition, but it should probably be published at some other venue. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 19:13, 24 August 2024 (UTC)
:::::I'm not married to using the tangent half-angle substitution instead of the tangent substitution. But it seems to me that the former is a lot simpler, and many of the basic properties of the trigonometric functions (especially their period) are more obvious.  Actually defining them via the tangent, one has to be careful about which sheet of the covering space one is on, because a periodic function of period 2&pi; is being described in terms of a periodic function of period &pi;. Hardy does this by continuation using a quarter period, but that's needlessly complicated. I'll try to track down a source that explicitly uses this approach. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 20:00, 24 August 2024 (UTC)
::::On this general theme you may enjoy
::::: {{cite arXiv |last=Robinson |first=Paul L. |year=2019 |title=A tangential approach to trigonometry |eprint=1902.03140}}
::::(though this is a self-published arXiv pdf, which probably doesn't count as a "reliable source" by Wikipedia standards). –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 19:20, 24 August 2024 (UTC)
I've gone ahead and just done Hardy's version.  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 14:00, 25 August 2024 (UTC)

:Is it really the case that {{tmath|1=\textstyle \tfrac12\pi = \int_0^\infty dx/(1+x^2)}} wasn't known until 1840? That seems impossible to me. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 18:05, 25 August 2024 (UTC)
::The text does not imply that. It says this was used as a ''definition'' of pi in 1841. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 18:47, 25 August 2024 (UTC)

== Concision ==

I am editing this article heavily for concision, including making the wikicode more concise. I strongly feel that concise wikicode looks prettier. Okay? [[User:Solomonfromfinland|Solomonfromfinland]] ([[User talk:Solomonfromfinland|talk]]) 14:51, 29 September 2024 (UTC)

:You are editing the style of the article (if you do this for consistency purposes, your choice has to be discussed here at the Talk page, it doesn't matter if "the code looks prettier"). Another thing I should discourage you from doing is changing "trigonometric" to "trig"; you can't do that; the word "trig" in this context is informal. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 14:57, 29 September 2024 (UTC)
::The second paragraf says, “The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent functions. Their reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions.”
::Could “in modern mathematics” be changed to “today”? After all, these functions are by definition part of math, so the word “mathematics” is redundant. When it says “…  cotangent functions”, could “functions” be removed, as it is obvious that these are functions? (More generally, can the word “function” sometimes be removed on the grounds that it is obvious that sine etc. are functions?)
::I admit, however, that i hav a habit of being aggressiv about editing articles for concision. [[User:Solomonfromfinland|Solomonfromfinland]] ([[User talk:Solomonfromfinland|talk]]) 15:11, 29 September 2024 (UTC)
:::"In modern mathematics" and "today" are not necessarily equivalent. "Modern mathematics" is quite old, actually. And yes, I agree with the removal of the word "function" when it appears later in the article. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 15:27, 29 September 2024 (UTC)
::::Thank you. [[User:Solomonfromfinland|Solomonfromfinland]] ([[User talk:Solomonfromfinland|talk]]) 15:29, 29 September 2024 (UTC)
::The section “'''Right-angled triangle definitions'''”, begins, “If the acute angle θ is given, then any right triangles with an angle of θ are similar to each other.” How about, “Any two right triangles with the same acute angle θ,  are similar to each other.” (I include a comma after a long subject of a sentence, such as here, for legibility.) Also, could “to each other” be removed?
::The next sentence is, “This means that the ratio of any two side lengths depends only on θ.” Could “This means that”, be replaced with “Therefore” or “Thus” or “That is,”?
::The next sentence is, “Thus these six ratios define six functions of θ, which are the trigonometric functions.” This is a bit wordy, since the word “function” is repeated. How about “Thus these six ratios define the six trigonometric functions of θ.”?
::Also, in said section, there is a long gap between the table, “Summary of relationships between trigonometric functions”, and the text above it; due to the pictures and their captions. This makes me want to shorten the captions. One caption says,
::“Top: Trigonometric function sin θ for selected angles θ, π − θ, π + θ, and 2π − θ in the four quadrants.
::Bottom: Graph of sine function versus angle. Angles from the top panel are identified.”
::In the expression “Trigonometric function sin θ”, i think “Trigonometric function” is redundant (kind of like “in color” in “red in color”); anyone who knows what sine is, knows that it is a trig function. Also, in “sine function versus angle”, “function” is redundant. [[User:Solomonfromfinland|Solomonfromfinland]] ([[User talk:Solomonfromfinland|talk]]) 15:28, 29 September 2024 (UTC)
:::Several parts of this article might benefit from a substantial rewrite, but trying to cut out random words from its sentences doesn't seem that helpful. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 15:31, 29 September 2024 (UTC)
::::What parts of the article should be rewritten, and how? Also, what do you mean by “random words”? Also, in the expression “Trigonometric function sin θ”, would you agree that “Trigonometric function” is redundant, at least in said caption? [[User:Solomonfromfinland|Solomonfromfinland]] ([[User talk:Solomonfromfinland|talk]]) 15:36, 29 September 2024 (UTC)
:::::For example, some subsections of {{alink|Definitions in analysis}} are too long and belabor the details. The section about {{alink|Radians versus degrees}} is too long and is distracting at this length near the top. The historical definition used for 2000 years between the 2nd century BC and at least the end of the 18th century, of "trigonometric lines" representing line segments with respect to a circular arc, is not discussed at all and is only mentioned in a half sentence. There is one sentence about "The sine and cosine functions are one-dimensional projections of uniform circular motion" but this should be a whole section. There is no discussion of the relation of trigonometric functions to vectors, but this should also be a whole section. There is one example of a Fourier series, but this should again be a whole section. The section {{alink| In the complex plane}} should be moved to top level and substantially expanded. The history section mentions "Historically, trigonometric functions were often combined with logarithms in compound functions like the logarithmic sine, ...", but this topic should again be a whole section (and ideally link to a main article which would discuss the topic in detail, since no Wikipedia article currently does). There should be a whole section about the relation between circular and hyperbolic functions, but currently the latter are barely mentioned here. The {{alink|Applications}} section is a mess in my opinion: the material currently there is basic tools of trigonometry and should be moved to separate top-level sections, but we don't mention any of the applications of trigonometry (historically astronomy, navigation, surveying, architecture, gunnery, ..., or more recently nearly every part of science and engineering).
:::::Anyway as for your specific questions: The sentences about similar triangles should be more substantially rewritten for clarity. I don't think your proposed changes are much better than what was there. {{tq|i=yes|"long gap between the table"}} – the images to the right there are not great and could probably be usefully moved or replaced, and the table's content also doesn't match the section, and it should be moved or modified. Shortening the captions doesn't really make a significant improvement in my opinion. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 17:13, 29 September 2024 (UTC)
::::::I edited the section "Radians versus degrees", lightly, for concision. "the images to the right there are not great and could probably be usefully moved or replaced, and the table's content also doesn't match the section, and it should be moved or modified." How should said images be moved or replaced? How should the table be modified or replaced? [[User:Solomonfromfinland|Solomonfromfinland]] ([[User talk:Solomonfromfinland|talk]]) 00:12, 30 September 2024 (UTC)
::::::“'''In the complex plane'''” is a subsection of section “'''Definitions in analysis'''”. Do you mean that “'''In the complex plane'''” should be moved so that it is in “'''Definitions in analysis'''”, right after the section header? Also, the section “'''Definitions in analysis'''” is rather long. Should it be edited for concision and if so, how? [[User:Solomonfromfinland|Solomonfromfinland]] ([[User talk:Solomonfromfinland|talk]]) 00:17, 30 September 2024 (UTC)
:::::::No, the complex plane section should be moved to the top level of the article, not made a subsection of "definitions in analysis". –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 02:35, 30 September 2024 (UTC)
::::::::You mean, “In the complex plane” should be a first-order section, not a subsection, and should should be the first thing in the article, other than the intro? [[User:Solomonfromfinland|Solomonfromfinland]] ([[User talk:Solomonfromfinland|talk]]) 11:49, 30 September 2024 (UTC)
:::::::::: This is my read, and parts of "Definitions in analysis" could be reorganized into that section, like Euler's formula, with Bourbaki's definition left in the analysis section. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 13:10, 30 September 2024 (UTC)
:::::::::No, it should clearly not be the first thing in the article. It belongs in the second half of the page. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 15:04, 30 September 2024 (UTC)

== Are they "discovered" ==

Parts of the article refer to the historical "discovery" of various trigonometric functions.  Is it better to say something along the lines of "first defined by"?  To me "discovery" means that the functions already existed and then were found, but I think of mathematical concepts as useful definitions rather than fundamentally already existing.  For example, is it possible to discover the function "{{math|1=''Q''uantling(''x'') = tan(''x'') − 0.193000669241688266320}}", or am I the first to have defined it? —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 15:19, 14 February 2025 (UTC)

:For example, in some fields of mathematics one can pick and choose among some axioms.  Seemingly these axioms are not fundamentally true in all contexts, ready to be discovered, but merely are useful in some contexts.  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 15:22, 14 February 2025 (UTC)
::{{ec}} This is a part of a longstanding debate (see [[Mathematical platonism]]) on reality of mathematics. I agree that for trigonometric functions, the right term is "defined". For axioms the right terms are "stated" in the case of an axiom modeling the reality, such as the [[Parallel postulate]] or "chosen" in the case of the specification of the theory (there are many axiom systems that define the same theory. In the case of a theorem or a conjecture, I would use "stated". It is only for proofs that I would use "discovered". However, this does not cover the cases of wide theories and branches of mathematics, such as [[set theory]], [[infinitesimal calculus]], [[Probability]] and [[distribution (mathematics)|distribution theory]] for which I would use "introduced" or "initiated". [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:49, 14 February 2025 (UTC)

: I would also prefer "defined" or "introduced" in many cases. (I'm fine with "discovered" a theorem.) However, sometimes I have seen "invented", which I think is even worse than "discovered". [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 15:24, 14 February 2025 (UTC)
::Thank you for the quick response.  Personally, I don't mind "invented" because, for me, it is pretty synonymous to "defined" in this context; but I will gladly go with "defined" if that is where the consensus is headed.  (I don't like "introduced" because, to me, it sounds like it existed before and is now known more widely.  The "now known more widely" is right, but the "existing before" doesn't sound right to me.)  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 15:32, 14 February 2025 (UTC)
::Yes, I agree that "discovering a proof" would be okay.  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 15:35, 14 February 2025 (UTC)
:It's usually better to not claim a "first" for specific characters, because such claims are nearly always wrong. The best is to just state exactly what we know: so-and-so specifically did such-and-such, and not speculate about whether or not anyone else had done so previously. One other phrasing that is usually accurate at the time (though can also become outdated) is something like "the earliest known X is found in Y" or "the oldest extant example of X is Y". –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 15:42, 14 February 2025 (UTC)
:I went ahead and boldly edited.  If that's wrong, please undo ... and discuss further here.  Thanks —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 15:51, 14 February 2025 (UTC)
:All of our articles [[Trigonometric functions]], [[Trigonometry]], and [[History of trigonometry]] also do a very poor job of explaining the varying mathematical context through history. There was no concept of "trigonometric functions" per se until quite recently. Through most of history these were taken to be line segments (and line segments were implicitly associated with their lengths), not numbers or ratios.
:It's anachronistic and misleading to say something like "X person invented the cotangent function" when an accurate description would be more like "X person wrote down a table associating the sun's altitude to the length of the shadow of a vertical gnomon (sundial rod), which 2 centuries later began to be used for more general trigonometric purposes. X's table was probably based on earlier examples, possibly much earlier, but they have been lost." –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 15:54, 14 February 2025 (UTC)

== Use of tau (𝜏) in addition to π ==

Could the "Algebraic values" section benefit from providing radian measures in terms of 𝜏 as well as π?  For people coming to trigonometry for the first time this could give an additional intuition linking angles to turns around the unit circle.

For example:

sin π/6 = sin 𝜏/12 = sin 30° = √1/2 = 1/2 [[User:TheGrifter80|TheGrifter80]] ([[User talk:TheGrifter80|talk]]) 02:28, 14 March 2025 (UTC)

: tau is fringe, so doesn't belong here. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 11:06, 14 March 2025 (UTC)
:IMHO, {{mvar|𝜏}} is interesting but not common enough in use for use in this article.  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 12:50, 14 March 2025 (UTC)
::Ok, fair enough [[User:TheGrifter80|TheGrifter80]] ([[User talk:TheGrifter80|talk]]) 09:36, 15 March 2025 (UTC)