Talk:Trigonometry

Relationships between angles and ratios of lengths

@Jacobolus: Here, in addition to being unsourced — which angles and which lengths? My index finger has a length, a road has a length, if I take their ratio is that trigonometry? Take the law of cosines, there's no simple "ratio of lengths" there. The sentence in the form introduced in January last year is pretty much meaningless. Ponor (talk) 15:22, 16 January 2024 (UTC)

This is not a controversial claim, and we can easily find hundreds of sources for it.

Trigonometry is most specifically about the metrical relations between intrinsic and extrinsic measurements between points on a circle, but more generally is about metrical relationships in the Euclidean plane ("planar trigonometry") or sphere ("spherical trigonometry"), or more generally still about metrical relationships in constant-curvature pseudo-Euclidean spaces. This includes not only solving triangles, but also things like the sums and differences of angles (if you like, you can interpret that as the analysis of various line segments related to cyclic polygons). The concepts first used for analyzing orbits of celestial bodies and metrical relationships in the plane are also are useful in a more abstracted setting for understanding and analyzing periodic functions, because those can be thought of as having the circle as their domain. –jacobolus (t) 15:34, 16 January 2024 (UTC)

I know, @Jacobolus, I know. But does my 15 year-old need to learn about "metrical relationships in constant-curvature pseudo-Euclidean spaces" to be able to understand what trigonometry is? Or about 99.8% other people? Find one good source, please, for a 15 year-old. Which angles, which lengths? The sentence doesn't even summarize the rest of the article. Check what goniometry is (for us here). Is Fourier analysis part of trigonometry too (ugh)?

Britannica has it like this: "Trigonometry, the branch of mathematics concerned with specific (trigonometric!) functions of angles and their application to calculations." Why can't we? Ponor (talk) 15:52, 16 January 2024 (UTC)

I understand your concerns. I rewrote the first sentence, hoping to resolve them. D.Lazard (talk) 16:25, 16 January 2024 (UTC)

@D.Lazard, thanks. I'm expaning an article on another wiki, and I think I'll go with "branch of mathematics that arose from the study of the relationship between the angles and side lengths of triangles." It kind-of says there's more to it, but doesn't go into "metrical relationships in constant-curv…". Ponor (talk) 16:38, 16 January 2024 (UTC)

That's not accurate either. It originally arose from the study of the relationship between arcs and chords of circles for use in spherical astronomy. –jacobolus (t) 16:53, 16 January 2024 (UTC)

Not an expert or science historian, but my impression from our article(s) and Britannica is that it was always about triangles: "Hipparchus was the first to construct a table of values for a trigonometric function. He considered every triangle—planar or spherical—as being inscribed in a circle, so that each side becomes a chord." Ponor (talk) 17:52, 16 January 2024 (UTC)

No, analyzing spherical triangles per se wasn't really a thing until at least a couple centuries later (you can find them analyzed in Menelaus's Spherics), and Hipparchus's astronomy was not based on triangles as a fundamentally important shape (obviously triangles appeared here, alongside other shapes). Even the early history of the "sine" and "cosine" functions was not about right triangles. The most important metrical relationship used in medieval Islamic spherical trigonometry was between segments in a complete quadrilateral. The "trigonometry" you are thinking of focused on solving triangles (and the name "trigonometry") is from more like the 16th century, but once solving triangles became a core part of the subject, it was still broader than that overall. Really the main thing that has distinguished what we now call "trigonometry" from other kinds of geometry, throughout its history, was the use of pre-computed tables relating chord (or sine, etc.) to arc or angle. If you solve a problem using a trigonometric table (or trig scale and dividers, or trig functions scales on a slide rule, or nowadays an electronic calculator/computer), then what you are doing is "trigonometry" whether or not it specifically involves a triangle. –jacobolus (t) 18:19, 16 January 2024 (UTC)

Actually, let me just quote Glen Van Brummelen's book The Mathematics of the Heavens and the Earth:

What Is Trigonometry? This deceptively difficult question will shape the opening chapter. The notion that sines, cosines, and other modern functions define what we mean by "trigonometry" may be laid to rest instantly; these functions did not reach their modern forms, as ratios of sides in right-angled triangles, until relatively recently. Even in their historical forms they did not appear until medieval India; the Greeks used the length of the chord of an arc of a circle as their only trigonometric function. The word itself, meaning "triangle measurement," provides little help: it is a sixteenth-century term, and much ancient and medieval trigonometry used circles and their arcs rather than triangles as their reference figures.

If one were to define trigonometry as a science, two necessary conditions would arise immediately:

• a standard quantitative measure of the inclination of one line to another; and
• the capacity for, and interest in, calculating the lengths of line segments.

We shall encounter sciences existing in the absence of one or the other of these; for instance, pyramid slope measurements from the Egyptian Rhind papyrus fail the first condition, while trigonometric propositions demonstrated in Euclid’s Elements (the Pythagorean Theorem, the Law of Cosines) fail the second.

What made trigonometry a discipline in its own right was the systematic ability to convert back and forth between measures of angles and of lengths. Occasional computations of such conversions might be signs of something better to come, but what really made trigonometry a new entity was the ability to take a given value of an angle and determine a corresponding length. Hipparchus’s work with chords in a circle is the first genuine instance of this, and we shall begin with him in chapter 2. However, episodes that come close—a prehistory—do exist in various forms before Hipparchus, and we shall mention some of them in this preliminary chapter. [...]

–jacobolus (t) 18:45, 16 January 2024 (UTC)

The summary "concerned with specific functions of angles and their application to calculations" is accurate, albeit very vague. (For what it's worth I don't think we should say much about non-Euclidean trigonometry in the lead section.) –jacobolus (t) 16:55, 16 January 2024 (UTC)

I think of geometry at the high school level as split roughly into 3 main approaches: "Greek" style as found in Euclid's Elements, with line segments considered per se (rather than measured) and compared using a compass, and with points, lines, and circles constructed using compass and straightedge; "trigonometry" style, from Babylonian astronomy, via Hipparchus and Ptolemy and Indian/Islamic-world mathematicians, with angles measured and line-segment lengths measured, the geometry of protractors and rulers; and "analytic" or "coordinate" style, from Fermat/Descartes (but with hints in Appolonius' Conics and in ancient cartography/astronomy), based on constructing a grid and then describing geometric objects using algebra. –jacobolus (t) 17:08, 16 January 2024 (UTC)

"[Trigonometry] originally arose from the study of the relationship between arcs and chords of circles for use in spherical astronomy": in astromomy arcs are measured as angles; chords are side lengths of some triangles (isoceles triangles). So, there is nothing inaccurate in saying that trigonometry is about the relationship between angles and side lengths of triangles. D.Lazard (talk) 18:46, 16 January 2024 (UTC)

It is inaccurate/anachronistic insofar as chords were not (at all!) conceived of as being about triangles, isosceles or otherwise. We've unfortunately lost the several ancient Greek books (by Hipparchus, Menelaus of Alexandria, and others) specifically focused on chords, but in the various applications still extant, the chord-related tools/theorems are not focused on triangles and typically don't explicitly make triangles at all. –jacobolus (t) 19:00, 16 January 2024 (UTC)

The word "trigonometry" is really an unfortunate misnomer. Saying "trigonometry is about triangles" is sort of like saying "calculus is about epsilon–delta proofs". It's not entirely wrong, but it is misleading. –jacobolus (t) 19:13, 16 January 2024 (UTC)

This discussion becomes silly. It started from the first sentence of the lead, and has derived to a discusion on the history of the subject, and on the historical meanings of "trigonometry". The first sentence must be about trigonometry in 2024. I do not see in the above discussion any argument against my version of the first sentence of the lead. Maybe, "more specifically" may be confusing, as suggesting that trigonometry consists only of the study of trigonometric functions. So, I have changed it to "in particular". Please, if you think that the fist sentence is not convenient, explain clearly the reasons, or provide an alternative version. D.Lazard (talk) 14:34, 17 January 2024 (UTC)

No I am talking about the (2024) extent of the subject of "trigonometry", not the "historical" meaning. Trigonometry is the study of the metrical relationships between distances and angles, and more generally the abstract study of tools developed for describing those relationships, including applications to more abstract situations which can be modeled using circles or periodic functions. Triangles are obviously relevant, but not the core point and not the extent of the subject. –jacobolus (t) 15:40, 17 January 2024 (UTC)

You're most likely right. But for the majority of readers, most likely very unfamiliar with trigonometry, that would explain nothing. One abstraction would be replaced with another abstraction. For most people trigonometry is about angles, triangles, and... trigonometric functions. Add a sentence saying that it's more than that (people can skip it if they don't understand it), but do not remove this basic notion.

"I could have done it in a much more complicated way," said the Red Queen, immensely proud. Ponor (talk) 16:11, 17 January 2024 (UTC)

What would explain nothing? Note that none of my comments here has been proposed as an alternative lead sentence. I'm talking to folks participating in this discussion here, not to some hypothetical "majority of readers". –jacobolus (t) 16:43, 17 January 2024 (UTC)

If I were going to rewrite this article (which would take quite a lot of work, and is something I don't intend to do today) I think the lead section would say something along the lines of:

Trigonometry is a branch of mathematics which studies the metrical relationships between angles and distances, characterized by the use of trigonometric functions. These functions, which relate an angle or arc of a circle to straight-line distances, can be thought of as describing the ratios between sides of a right-angled triangle or the Cartesian coordinates of a moving point on a circle, and are periodic (repeating) functions of angle.

Trigonometry emerged out of ancient Babylonian and Greek astronomy, then underwent significant development in medieval India and the Islamic world. The name trigonometry, from Greek roots τρίγωνον (trígōnon) 'triangle', and μέτρον (métron) 'measure', was coined in 16th century Europe, where solving triangles became a central focus of the subject. [...]

I'd have to put some more work in to make up a version I was really happy with though. –jacobolus (t) 17:11, 17 January 2024 (UTC)

"Allied angles" listed at Redirects for discussion

The redirect Allied angles 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 Wikipedia:Redirects for discussion/Log/2024 February 7 § Allied angles until a consensus is reached. Duckmather (talk) 20:04, 7 February 2024 (UTC)

Moving my comments from the redirect deletion discussion to here (jacobolus, 25 April):

This seems to be a term nowadays only used in India (e.g. Bharadwaj 1989). It can be found in some older English sources though (e.g. Hall & Knight 1893, Bowley 1913, Briggs & Bryan 1928). Looks like any pair of angles whose sum or difference is a multiple of 90° (π/2 radians) are considered "allied". Conceivably we could make a new article entitled Allied angles and redirect Supplementary angle, Complementary angle, etc. to there. I have long thought those should be their own article instead of a redirect to Angle.

Though there's apparently also a second meaning of "allied angles", which is consecutive interior angles ("co-interior angles") of a transversal; if the two lines transversed are parallel, two such angles are supplementary. (Example sources: Durell 1939, Hislop 1960). –jacobolus (t) 21:59, 7 February 2024 (UTC)

How to proof identies

How to proof identies Please help me @Pi 2 41.121.19.115 (talk) 13:44, 25 April 2024 (UTC)

This is a place for discussing how to improve this article about trigonometry, not a general help forum about trigonometry. I would recommend you try someplace like http://reddit.com/r/learnmath instead. –jacobolus (t) 14:42, 25 April 2024 (UTC)

"Polygonometry"

A user insists to add in the first paragraph of the lead a sentence about "polygonometry". I'll to revert them again for the following reasons.

• "Polygonometry" is not a commonly used word nor a known area of mathematics
• the link to generalized trigonometry is wrong (this article is about trigonometry in non-Euclidean spaces)
• The assertion that every polygon can be decomposed into triangles is true but has nothing to do here
• The formulation suggests wrongly that polygonometry is more important than trigonometry
• Computation of angles and side lengths of polygons is certainly an application of trigonometry, but it is not its heart.
• Over all: the first sentence is for a short description of the subject, not for its possible uses.

This said, I would not oppose to add to the third paragraph of the lead (the one about applications), a sentence saying that trigonometry can be used to compute angles and side lengths of every polygon. D.Lazard (talk) 15:02, 26 May 2024 (UTC)

@D.Lazard: some good points above, thank you. My priority here is to establish the core notability of trigonometry. Per WP:LEAD, a good lead section cultivates interest in reading on… in a clear, accessible style… [and should] establish context, [and] explain why the topic is notable.

At the moment the lead is missing the most important point - that trigonometry is not really just about triangles. It is the foundation for the study of all polygons.

Having said this, I would prefer to widen the point further, to explain why trigonometry is foundational to so many branches of mathematics. It all comes down to the point that triangles are the simplest polygon.

Onceinawhile (talk) 15:44, 26 May 2024 (UTC)

Please, do not present your opinion as facts: Trigonometry is not foundational for any branch of mathematics. It is a part of geometry that is used as a computational tool in many applications of geometry. Historically, the application of trigonometry that can be called fundamental (not foundational) is spherical geometry, and its applications to navigation and celestial mechanics. This is historically, as wells as nowadays, much more important than the applictions to geometry of polygons; this is the needs of these applications that motivated the invention of trigonometry. Another fundamental application of trigonometric functions is Fourier analysis. Clearly, polygons are not involved an any of these applications.

Again, do not present your opinion as facts, and the importance that you give to the study of polygons is only an opinion. D.Lazard (talk) 17:27, 26 May 2024 (UTC)

Your comment orders me twice to "not present [my] opinion as facts", whilst the main body of your comment is presenting your opinion as facts.

If your position is more correct than mine, you need to bring sources for it. So far you have provided no sources, whilst deleting three of my sources and quotes.

Onceinawhile (talk) 21:43, 26 May 2024 (UTC)

There is no problem to present in talk pages personal opinions on how present things in articles. I know that what I wrote is my opinion, and this is the reason for which I never try to add them on the aricle. But the existence of argumented opinions that challenge what you wrote in the article is strong indication that you tried to present your opinion as fact in the encyclopedia. D.Lazard (talk) 08:58, 27 May 2024 (UTC)

The name "trigonometry" is bit of a misnomer, as the subject of trigonometry is not primarily about measuring triangles (some people prefer the name "goniometry" which is more accurate), somewhat similar to the way "geometry" is not primarily about measuring the earth. Trigonometry's primary focus is relating circular arc lengths / angle measures to straight line segments (and using similarity, to abstracted line segments / coordinates based on a standard circle for which a trigonometric table or "trigonometric function" is defined, which can be dilated to fit any desired circle). As D.Lazard points out, the original application was to spherical geometry in the form of spherical trigonometry, for use in astronomy, but later also celestial navigation, geodesy/cartography, and much later also geology, crystallography, etc. Planar trigonometry was eventually used for optics, surveying, gunnery, architecture, trades, engineering, ..., but again while there were often triangles involved, the most fundamental feature was usually circles.

Solution of (right or general) triangles is one particular application of trigonometry which arose in medieval Islamic mathematics and then was further developed in Renaissance Europe. It's an important application which is especially central to teaching in high school courses, but is still not really the foundation of the subject.

In the past century or two, trigonometry is more generally the study of trigonometric functions, for which more advanced applications involve quite a bit of mathematical analysis. The "circles" which show up are a bit abstract, describing for instance the domain of arbitrary periodic functions, which makes trigonometry important to the study of differential equations, signal processing, etc.

"Polygonometry", meaning the study of metrical relations of polygons including angle measures, is a relatively niche topic dating from the 18th century and rarely presented systematically. Polygons are instead studied from a variety of points of view in various applications. The application which is closest to what might be called "polygonometry" is probably the kinematics of linkages. You could also consider surveying to involve a lot of "polygonometry" but it's not really thought of that way as far as I can tell. In computational geometry there is significant study of polygons, but usually based on analytic geometry and vectors rather than angle measures. –jacobolus (t) 07:55, 27 May 2024 (UTC)

Thank you jacobolus. This is a helpful build - per above we need to make clear in the lead that "trigonometry is not really just about triangles". I will try to bring a few more sources from differentt perspectives. Onceinawhile (talk) 08:16, 27 May 2024 (UTC)

@D.Lazard and Jacobolus: I have taken the time to read this talk page more broadly, particularly the core definition. I see that in January there was a debate about this edit which then petered out. This is the same problem that triggered me to edit here in the first place. It is simply misleading to suggest that trigonometry is just about triangles. The Glen Van Brummelen quote that jacobolus provided sets it out nicely (see here for convenience, and the "What's in a name?" section in volume 2). Since Van Brummelen's two volumes in the Princeton University Press represent the most detailed broad-concept study of the topic that has been published, our article should follow his explanation.

As an aside, for an illustration of how much our readers would benefit from a clear and well-sourced dispelling of the myth that trigonometry is just about triangles, look how frequently this same issue comes up elsewhere on the internet ([1][2][3][4]).

Onceinawhile (talk) 16:32, 1 June 2024 (UTC)

Yeah, I would like to eventually do a substantial rewrite of this page. It's just a bit of a daunting project that takes finding some substantial time and a bit of pep talk to get motivated to take on. It's a lot easier to respond to occasional talk page discussions. :-) –jacobolus (t) 16:54, 1 June 2024 (UTC)

Overlap

There seems to be a substantial overlap between Trigonometry (as the study of trigonometric functions) and Trigonometric functions, the former of which I just translated, which made me wonder what should go where. Does the "common trigonometric values" table belong here? Mnemonics? Does the "inverse trigonometric functions" section belong to Trigonometric functions? Should more be said about Fourier analysis here? Differential equations related to trig. functions? I understand no one is quite happy with this article (@Jacobolus!), and also that no one has time for a good rewrite, so how about we do some pruning here and there, so there's at least a little less overlap? There's this m:List of articles every Wikipedia should have/Expanded/Mathematics that lists both articles as (somewhat) vital, but no wiki, from what I can tell, has a clear separation between the two. Thoughts? Ponor (talk) 16:13, 27 June 2024 (UTC)

Substantial overlap is not a problem, as long as two articles don't have the same scope. This article trigonometry should be a much expanded high-level overview rather than a reference work, and should definitely discuss more about Fourier analysis, differential equations, etc. Overall what this topic needs is better sources and significant expansion, especially of prose, not pruning.

We should aim for complete and reasonably self contained separate articles for, among others, angle (currently a mess), angle measure (redirects), unit circle (incomplete and mediocre), history of trigonometry (incomplete and mediocre), trigonometric table (incomplete and mediocre), trigonometric functions (far too formula heavy), trigonometric scale (or similar title; we currently have the start of an article scale of chords, so it could be moved and expanded), logarithmic trigonometric functions (no current coverage of the topic), chord (trigonometry) (redirects), sine, tangent (trigonometry) (redirects), secant (trigonometry) (redirects), versine (needs a rewrite), half-tangent (i.e. the tangent of half an angle, red link), exsecant (I recently cleaned this one up), inverse trigonometric functions (not accessible enough), possibly arcsine and arctangent (both redirect), solution of triangles (needs more sources and historical discussion), ...

And then we should have separate articles for all of the major theorems and identities. Law of cosines is passable but needs sources for proofs. Law of sines needs work. They generally get weaker from there. Pretty much every section at list of trigonometric identities, e.g. angle sum identity, should be backed by a separate article. The article Proofs of trigonometric identities should be deleted and any salvageable content merged into articles about each separate topic.

Our article about spherical trigonometry is a complete mess. We should have separate articles about spherical triangle (redirects), spherical excess (redirects), polar triangle (redirects), polar duality (red link), Girard's theorem (redirects), Napier's analogies (redirects), Delambre's analogies (redirects), ... –jacobolus (t) 17:29, 27 June 2024 (UTC)

Hyperbolic functions?

Trigonometry is the study of circular functions; is there an analogous name for the stufy of hyperbolic functions? I'd like to add a {{distinguish}} hatnote if a suitable article exists. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:18, 22 August 2024 (UTC)

No, but you could consider it to be part of trigonometry if you want, and add a section down near the bottom of this article. –jacobolus (t) 15:46, 22 August 2024 (UTC)

Spherical trigonometry

I'm considering adding {{about|triangles in the Euclidean plane|triangles on a sphere|Spherical trigonometry|the album|Trigonometry (album)|the TV series|Trigonometry (TV series)}}. Is that enough, or does the article need a new section with a {{main}} template? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:23, 2 September 2024 (UTC)

Per WP:RELATED, this isn't the intended use of hatnotes. Instead, the lead section of this article should mention Spherical trigonometry with a link, and it should ideally include a dedicated section with a {{main}} link. –jacobolus (t) 18:10, 2 September 2024 (UTC)

Semi-protected edit request on 14 December 2024

REMOVE the link from the word 'adjacent' in the definition of Cosine and instead CHANGE it to an italicisation of it so it connotes that the word is being turned into a standard trigonometric term. Ideally, you could add a paragraph *before* the trigonometric definitions where the words 'opposite' 'adjacent' and 'hypotenuse' are defined. For a person not versed in trig, the word 'adjacent' could be either of the two sides that make the angle and the current phrasing is not pedagogically explicit that a standardized term is being introduced. Regardless, the hyperlink needs to be removed because it takes you to a useless target. The current link takes you to an article that has NO mention of the word 'adjacent' as used in trigonometric ratios. Thanks. 24.161.66.164 (talk) 18:05, 14 December 2024 (UTC)

 Done I moved the paragraph with term definitions in front of the trig definitions and removed the redirected link to triangle from 'adjacent'. I did not take the step of changing bolded font to italics, and have no preference if another editor chooses to do so. DrOrinScrivello (talk) 20:57, 14 December 2024 (UTC)