Talk:Trigonometric functions: Difference between revisions

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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)

== 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

:<math>\sin \frac\pi3 = \sin 60^\circ = \frac{\sqrt{3}}{2} = \frac{1}{2}</math>

which is not the case. It's correct in the table next to the diagram.

[[User:Mwasheim|Mwasheim]] ([[User talk:Mwasheim|talk]]) 14:55, 9 April 2023 (UTC)

:Sorry, in my understanding it says:

:<math>(\cos,\sin)(\frac{\pi}3) = (\cos,\sin)(60^\circ) = (\frac{1}{2},\frac{\sqrt{3}}{2})</math>,

:which is correct. –[[User:Nomen4Omen|Nomen4Omen]] ([[User talk: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! [[User:Mwasheim|Mwasheim]] ([[User talk: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 ==

@[[User:D.Lazard|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? —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 13:14, 29 September 2023 (UTC)

:This is discussed at [[Mnemonics in trigonometry]]. You can navigate to [[SOHCAHTOA]] to see the relevant section. –[[user:jacobolus|jacobolus]] [[User_talk: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? [[User:Sniperstock9041|Snipe]] ([[User talk:Sniperstock9041|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. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 05:37, 23 November 2023 (UTC)

== Notes and References ==

Line 145 ⟶ 100:

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)

== Degree sign explicitly shown ==

In the [[Trigonometric functions#Radians versus degrees|Radians versus degrees]] section, it says: "If units of degrees are intended, the degree sign must be explicitly shown (sin x°, cos x°, etc.)." I don't think this makes any sense. Extracting the unit out of a variable (i.e. making the variable a dimensionless scalar) is discouraged pretty much everywhere, as far as I'm aware. (In this case the degree is not really even a unit but more like a constant, which makes this even worse.) I'm going to remove the passage. [[User:Granite11|Granite11]] ([[User talk:Granite11|talk]]) 20:11, 1 July 2025 (UTC)

:Let's discuss this first before editing. —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 20:15, 1 July 2025 (UTC)

:I think of the degree sign not as a {{!tq|unit}} but as a {{tq|multiplicative factor}}, with the value {{math|''π''/180}}. That is, much as {{math|90''x''}} means {{math|90}} times whatever value {{mvar|x}} has, an expression like {{math|90°}} means {{math|90}} times {{math|''π''/180}}, which is {{math|''π''/2}}, and is thus converted to radians before being handed to a trigonometric function. —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 20:20, 1 July 2025 (UTC)

::I just published my changes before reading your reply—sorry about that. I think all of the text I removed was either redundant, or in the case of the claim that a degree sign must be suffixed to the variable symbol, sketchy. I agree completely with your view of the degree as a multiplicative factor, and the last sentence of the section mentions that still. [[User:Granite11|Granite11]] ([[User talk:Granite11|talk]]) 21:59, 1 July 2025 (UTC)

:This doesn't make sense to me either. I would never write {{tmath|\cos x^\circ}}, and can't remember having ever seen that notation. It's just {{tmath|\cos x}}, even if {{tmath|x}} is an angle that has been specified in degrees.

:What is fair to say is that someone should try not to write {{tmath|\cos 60}} when they mean {{tmath|\cos 60^\circ}}, because the former is potentially confusing. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 03:04, 2 July 2025 (UTC)

:: For what it's worth, in mathematica the degree symbol is a constant numerical factor. This is potentially a source for this sometimes being the case. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 11:58, 2 July 2025 (UTC)