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	Trigonometric functions is a former featured article. Please see the links under Article milestones below for its original nomination page (for older articles, check the nomination archive) and why it was removed.
	This article appeared on Misplaced Pages's Main Page as Today's featured article on March 6, 2004.
Article milestones Date Process Result December 12, 2003 Featured article candidate Promoted September 20, 2004 Featured article review Kept July 19, 2008 Featured article review Demoted
Current status: Former featured article

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Algebraic value of sin 45°

@DVdm: reverted my change from

${\displaystyle \sin {\frac {\pi }{4}}=\sin 45^{\circ }={\frac {\sqrt {2}}{2}}={\frac {\sqrt {2}}{2}}}$

to

${\displaystyle \sin {\frac {\pi }{4}}=\sin 45^{\circ }={\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}}$

with the comment "Sqrt(2)/2 is much more common than 1/sqrt(2) in the literature". Though I understand that √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,
cmɢʟee⎆τaʟκ 22:55, 5 July 2020 (UTC)

Crissov added the accidentally duplicate values on 23 January 2020. Previously, the values looked like this and the duplication probably came from the "easy way to remember" values. Probably best to omit. Johnuniq (talk) 23:57, 5 July 2020 (UTC)

Yes, I already removed the duplicate values: . - DVdm (talk) 06:33, 6 July 2020 (UTC)

It might make sense to add a note to show it either way Bera678 (talk) 12:42, 17 December 2023 (UTC)

Unit Circle Diagram

Contains a number of errors. I'm not in a good position (ie. edit SVGs) to correct them. The value for sixty degrees is given as

${\displaystyle \sin {\frac {\pi }{3}}=\sin 60^{\circ }={\frac {\sqrt {3}}{2}}={\frac {1}{2}}}$

which is not the case. It's correct in the table next to the diagram. Mwasheim (talk) 14:55, 9 April 2023 (UTC)

Sorry, in my understanding it says:

${\displaystyle (\cos ,\sin )({\frac {\pi }{3}})=(\cos ,\sin )(60^{\circ })=({\frac {1}{2}},{\frac {\sqrt {3}}{2}})}$,

which is correct. –Nomen4Omen (talk) 15:08, 9 April 2023 (UTC)

Um, that is correct. Hmm. I was busy converting between radians and degrees and didn't read the legend carefully enough. Thanks! Mwasheim (talk) 15:21, 9 April 2023 (UTC)

as an aside, I wonder if it would help people sitting there with calculators which use radians (ie. all?) to specify that sin 90 is generally sin(rad(90)) on calculators?

SOHCAHTOA

@D.Lazard et al., I have heard many people talk about "SOHCAHTOA". Although I don't have a citation at hand, I think it is improvement to the article to discuss SOHCAHTOA. Hopefully, someone will come up with a citation pronto and this will all be moot but, even if not, might we keep this discussion anyway? —Quantling (talk | contribs) 13:14, 29 September 2023 (UTC)

This is discussed at Mnemonics in trigonometry. You can navigate to SOHCAHTOA to see the relevant section. –jacobolus (t) 13:48, 29 September 2023 (UTC)

Merging pages

I was on this page and I became curious: why do tangent, cotangent, secant and cosecant all share a page while sine and cosine get their own? Why can't we move them all into one page? Snipe (talk) 01:16, 23 November 2023 (UTC)

These were merged (a long time ago) because it was thought that most relevant material about any of them would be applicable to all of the others, many relevant features would be useful to compare from one to another, and important context was needed in common between them. People thought separate articles would consist of substantially duplicated material.

In retrospect I think this was the wrong decision (or rather, I think we should have an article called trigonometric functions about some common features and also have separate articles). Having these lumped together has discouraged people from adding useful information which applies to only one or another. Any subject which has enough to say about it independently to flesh out a self-contained article should generally have one, and there is quite a lot to separately say about sine, tangent, secant, etc., especially discussion about history. I eventually intend to make separate tangent (trigonometry) and secant (trigonometry) pages, and would be opposed to getting rid of the sine and cosine page, which should in my opinion also be expanded and somewhat reorganized. –jacobolus (t) 05:37, 23 November 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 Bera678 (talk) 14:01, 17 December 2023 (UTC)

There are a variety of reference styles used in Misplaced Pages 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. –jacobolus (t) 16:09, 17 December 2023 (UTC)

But this prevents adding notes to the article? 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. –jacobolus (t) 16:51, 17 December 2023 (UTC)

To add notes to the article, the {{notelist}} command must be used in the notes section. 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 <ref>...</ref>. –jacobolus (t) 09:11, 19 December 2023 (UTC)

"Logarithmic sine" listed at Redirects for discussion

The redirect Logarithmic sine has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Misplaced Pages:Redirects for discussion/Log/2024 January 29 § Logarithmic sine until a consensus is reached. 1234qwer1234qwer4 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. Voproshatel (talk) 19:01, 4 February 2024 (UTC)

yeah i also kind of agree User: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. –jacobolus (t) 01:43, 17 April 2024 (UTC)

Oh, ok User:Hamterous1 (discuss anything!🐹✈️) 11:50, 17 April 2024 (UTC)

Offensive content

I am astounded that Misplaced Pages 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.71.31.145.237 (talk) 11:29, 21 August 2024 (UTC)

Misplaced Pages does not insult by saying that all science teachers are crazy. It reports (with a reliable source) that some people use that mnemonic in trigonometry. Just like Misplaced Pages does not say that Trump is a liar. It gives a list of reliably documented false or misleading statements by Donald Trump. - 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! —Quantling (talk | 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.) –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. XOR'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. Dolphin (t) 03:15, 22 August 2024 (UTC)

Bartle and Sherbert

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

3rd edition, page 247. Tito Omburo (talk) 19:58, 23 August 2024 (UTC)

OK, added. XOR'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. Tito Omburo (talk) 14:05, 25 August 2024 (UTC)

"Definition via integration" section

@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 Misplaced Pages. It will take a lot of work, especially trimming, to salvage some parts as a Misplaced Pages 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. –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. Tito Omburo (talk) 18:00, 24 August 2024 (UTC)

I think the section as currently stands violates the spirit of both Misplaced Pages:No original research and Misplaced Pages: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. –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. 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. –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 2π is being described in terms of a periodic function of period π. 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. Tito Omburo (talk) 20:00, 24 August 2024 (UTC)

On this general theme you may enjoy

Robinson, Paul L. (2019). "A tangential approach to trigonometry". arXiv:1902.03140.

(though this is a self-published arXiv pdf, which probably doesn't count as a "reliable source" by Misplaced Pages standards). –jacobolus (t) 19:20, 24 August 2024 (UTC)

I've gone ahead and just done Hardy's version. Tito Omburo (talk) 14:00, 25 August 2024 (UTC)

Is it really the case that ⁠${\displaystyle \textstyle {\tfrac {1}{2}}\pi =\int _{0}^{\infty }dx/(1+x^{2})}$⁠ wasn't known until 1840? That seems impossible to me. –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. 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? 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. 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. 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. A1E6 (talk) 15:27, 29 September 2024 (UTC)

Thank you. 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. 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. –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? Solomonfromfinland (talk) 15:36, 29 September 2024 (UTC)

For example, some subsections of § Definitions in analysis are too long and belabor the details. The section about § 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 § 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 Misplaced Pages 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 § 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. "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. –jacobolus (t) 17:13, 29 September 2024 (UTC)

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