Talk:Trigonometry/Archive 2

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Difference between trigonometric and analytic definitions of "sine"

The high school definition of trigonometric function "sine" as dimensionless ratio of opposite side to hypotenuse inside a right triangle is curvature-dependent, and is not equivalent to the analytic definition of the "sine" function using Taylor series, which is curvature-independent.

Only in flat space, the trigonometric definition of "sine" as ratio of sides

$\sin \alpha :={\frac {a}{c}}$

is equivalent to the analytic definition of the "sine" using Taylor series:

$\sin \alpha :=\alpha -{\frac {\alpha ^{3}}{3!}}+{\frac {\alpha ^{5}}{5!}}-{\frac {\alpha ^{7}}{7!}}+\ldots$

In curved space, the two definitions produce different quantities, so they are not equivalent.

Why these two definitions are not equivalent?

A sphere with a spherical triangle on it.

Because the geometric expression ${\frac {a}{c}}$ is curvature-dependent, whereas the algebraic expression $\alpha -{\frac {\alpha ^{3}}{3!}}+{\frac {\alpha ^{5}}{5!}}-{\frac {\alpha ^{7}}{7!}}+\ldots$ is curvature-independent. A visual way to see it, is to draw a right triangle on a sphere and notice that the ratio of the side $a$ and hypotenuse $c$ has nothing to do with the analytic "sine".

How to verify that these two definitions are not equivalent?

Easy with dimensional analysis: the side length $a(m)$ has units of meters, and the hypotenuse $c(m)$ has units of meters, so

$\sin \alpha :={\frac {a(m)}{c(m)}}$

In contrast, in the analytic expression $\alpha (1)$ is dimensionless quantity, i.e., pure number

$\sin \alpha :=\alpha (1)-{\frac {\alpha (1)^{3}}{3!}}+{\frac {\alpha (1)^{5}}{5!}}-{\frac {\alpha (1)^{7}}{7!}}+\ldots$

The above implies that only the expression in which "meters" appear is curvature-dependent. Thus, "trigonometric proofs" of the Pythagorean theorem are not "circular" as long as they use the trigonometric definition of "sine" as ratio of two sides.

Alternative way to see that "there is a difference" in the two definitions is to count the number of arguments: the trigonometric definition of "sine" is a multivariable function and has two arguments $f(a,c)$, whereas the analytic expression is a single-variable function and has only one argument $f(\alpha )$.

Why should Wikipedia care?

Because the article should be accessible to high school students, and because when these students ask questions on math.stackexchange about the possibility of trigonometric proof of the Pythagorean theorem, they get incorrect answers that the trigonometric functions are defined by Taylor series, hence the proofs of the Pythagorean theorem are circular. Also, the "professional mathematicians" at math.stackexchange immediately troll and down vote correct answers, which point to their erroneous claim.

In summary, mentioning that the two definitions are equivalent only for flat space is NOT a pedantic insertion by me so that @user:jacobolus can delete it as "unhelpfully pedantic. in spherical/hyperbolic trigonometry no one ever takes ratios of angle-measure- or intrinsic-length-valued sides".

Keep in mind that Wikipedia is NOT intended solely for reading by persons who already have PhD in mathematics. The general reader would like to know that the trigonometric ratio definition does not work for curved space. After all, remember, he is general reader and does not know what mathematicians do or do not do, until the article says so.

In summary, what I expect? I expect that @user:jacobolus self-reverts his revert of my edit. Danko Georgiev (talk) 11:09, 8 April 2023 (UTC)

Your edit gives a misleading impression of the history/practice of trigonometry, which is at best going to go unhelpfully over the heads of the readers you are worried about, if it doesn’t confuse them. There is no such thing as “trigonometric definitions of sine and cosine” in curved space, the way your additions to Pythagorean theorem imply. Spherical trigonometry has a wide variety of formulas, none of which involve directly dividing the sides (for a triangle ABC on a sphere with center O, the "sides" are the measures of the central angles a = ∠BOC, etc.).

incorrect answers that the trigonometric functions are defined by Taylor series – this is a matter of convention (in this case, the convention followed by most modern advanced mathematics books). The traditional definition (from ~200 BC until ~1700) of the sine, cosine, tangent, secant, etc. was particular line segments associated with a circular arc. These originally arose in the context of spherical trigonometry (part of astronomy), no flat triangles in sight. –jacobolus (t) 14:46, 8 April 2023 (UTC)

Your reply is deeply confused, as evident from "There is no such thing as “trigonometric definitions of sine and cosine” in curved space". You seem not to be able to comprehend that a Definition contains ALL THE INFORMATION IN ITSELF. You do not have to mention any history in order to discuss the following

Definition: "sine is the ratio of opposite side to hypotenuse in right triangle".

The definition does not say whether the right triangle is drawn "inside flat space" or "inside curved space". It is exactly because the definition is silent on the curvature of the space that it is not equivalent to the analytic Taylor series. If the definition DOES NOT SAY ANYTHING about the curvature, you can apply the given definition either in flat space or in curved space. In fact, if you do not specify that you are drawing the "right triangle" inside a flat space, all of the formulas given as relation to the analytic functions will be FALSE. Danko Georgiev (talk) 15:22, 8 April 2023 (UTC)

This definition is never used outside the context of plane geometry (in particular, introductory high-school-level textbooks of the past century or two?), and implying that it is is misleadingly ahistorical. –jacobolus (t) 15:51, 8 April 2023 (UTC)

If you do not say what the context is, e.g. you wrote "the context is plane geometry", everything written in the article will be false. You cannot assume that a reader who does not know what is "the context" will guess what you could have written explicitly but did not write because you expected the reader to telepathically know. By the way, when you say "the context is plane geometry", I have no idea why you changed the word "flat geometry" to "plane geometry". "Plane" by definition is a two-dimensional surface. Are you talking about "flat plane" or about a "curved plane"? Do you by chance want to say that "the context is FLAT plane geometry"? Danko Georgiev (talk) 16:01, 8 April 2023 (UTC)

You seem to have non-standard foundational conceptual understanding and non-standard definitions of basic terms. That’s fine – for yourself – but before imposing those on heavily viewed Wikipedia articles, you must find corroborating reliable sources and consider WP:UNDUE and WP:FRINGE. Wikipedia is intended to reflect the consensus of mainstream published works, and is not the best venue for exploring alternative definitions unless they are supported by published sources. Edit: to elaborate, the “plane” in “plane geometry” or “planimetry” refers to the so-called Euclidean plane; the Latin “planum” and the French “plan” from which the English term arises literally mean “flat surface”. Both this and the English word “flat” ultimately descend from the same Greek root, πλατύς. ––jacobolus (t) 16:17, 8 April 2023 (UTC)

Did people take into consideration Euclid's 5th parallel postulate when they invented the word "plane"? Doing linguistic analysis on the origin of a word is NOT a valid mathematical method of proving theorems. Danko Georgiev (talk) 17:04, 8 April 2023 (UTC)

When mathematicians or scientists use the unqualified word “plane” (both historically and today) they almsot always mean the Euclidean plane, except in contexts where some other kind of plane is obviously more relevant. –jacobolus (t) 18:04, 8 April 2023 (UTC)

If you want an analogous spherical-geometry definition of the sine of an angle (dihedral angle) in a right-angled triangle, you need to take the ratio of the sine of the opposite side (central angle) to the sine of the hypotenuse. ––jacobolus (t) 15:55, 8 April 2023 (UTC)

You are changing the subject. I do not want analogous spherical-geometry definition. Instead, I am using exactly the definition as it is written in the main text. The main text does not introduce that the context is "plane geometry". In fact, I am using the term "flat space" because the issue is the "flatness". Is the space "flat" or is it "curved"? When you say "the context is plane geometry", I have no idea why you changed the word "flat geometry" to "plane geometry". "Plane" by definition is a two-dimensional surface. Are you talking about "flat plane" or about a "curved plane"? Do you by chance want to say that "the context is FLAT plane geometry"? Danko Georgiev (talk) 16:07, 8 April 2023 (UTC)

An unqualified “triangle” in the context of this article (and every other ordinary context in mathematics, unless otherwise specified) means a triangle in Euclidean space (sometimes an unqualified “triangle” might be taken to be in affine or projective space, but in those contexts there is no concept of length or angle measure, so they don’t involve trigonometry in the sense of this article). An unqualified “right triangle” is a triangle in Euclidean space with one right-angled corner. If you want to talk about spherical right triangles or hyperbolic right triangles or Lorenzian right triangles or what have you, you need to specify that or explicitly put yourself into a context where those are the obvious subject. –jacobolus (t) 16:33, 8 April 2023 (UTC)

You are confusing "a system of axioms" with the "semantics (i.e. meaning) of a particular axiom". In absolute geometry you have the first 4 postulates of Euclid, but not the Euclid's 5th parallel postulate. In absolute geometry, you can talk about lines, planes and triangles, but you do not have "flatness" as concept, which is introduced by the Euclid's 5th parallel postulate. What you are just saying, when translated in plain English is that when you say the word unqualified “triangle” it already contains as implicit meaning all 5 postulates of Euclid. So only with you, I cannot possibly talk about a definition of "sine" or a particular definition of "triangle" because you already have in "your English" that each of these words already contain as implicit meaning all 5 postulates of Euclid !!! So if you put all 5 postulates of Euclid, in each and every word of yours, I am afraid that there is nothing more to discuss with you. Indeed, I do not know how a person can ever talk with you about each of the Euclid's postulates, separately, one by one? I have done my fair effort to communicate with you, so that you understand me. However, you never tried to establish two-way communication. OK, fine. I no longer want to correct anything in this article. Have a nice day! Danko Georgiev (talk) 16:48, 8 April 2023 (UTC)

Yes that is correct, plane trigonometry as a subject of study, about which you can find thousands of books from the past several centuries (as distinct from spherical trigonometry, a separate subject about which you can also find thousands of books; sometimes the two subjects are combined as two parts of a single volume) is built on top of Euclidean planar geometry (which had Euclid's Elements as the canonical source). This is not something I am inventing: you can go read these books for yourself. (If you would like I can provide a few dozen references to some of the more popular titles from various centuries.)

If you want to make the context more explicit in this article that is fine with me. But implying that the concept of "sine" as the ratio of sides of a right-angled triangle still applies in curved space runs contrary to centuries of convention and is confusing/misleading to readers. –jacobolus (t) 17:54, 8 April 2023 (UTC)

You may find MacFarlane (1893) "On the Definitions of the Trigonometric Functions" of some interest. ––jacobolus (t) 19:40, 8 April 2023 (UTC)

I agree mostly with jacobolus' arguments. However, his revert of Danko Georgiev's edit is fully justified for another reason: "trigonometric ratios" are defined earlier in this article, and anything that suggests another definition is definitively confusing. If one would desire to emphasize that only Euclidean geometry is used in this article, this should be done much earlier. Personally, I would oppose to such a mention, because I cannot imagine a reader who does not know basic trigonometry and knows non-Euclidean geometries. D.Lazard (talk) 18:49, 8 April 2023 (UTC)

Is it allowed?

Hello, I want to copy these contents and then edit them. Do I have permission? Germany Poul Ah (talk) 04:02, 13 June 2023 (UTC)

You are going to need to be more specific. Wikipedia article text is licensed under the Creative Commons Attribution-ShareAlike License 4.0. You are allowed to make derivative works, but if you distribute them it must be under the same license, which grants readers the same right to make and redistribute their own derivative works while requiring attribution of the original author(s). –jacobolus (t) 05:06, 13 June 2023 (UTC)

This is not the case. Can I use it as a source and not as a copyright? Germany Poul Ah (talk) 14:08, 13 June 2023 (UTC)

I don't understand your question, sorry. –jacobolus (t) 15:52, 13 June 2023 (UTC)

I say, as the author of the supplementary mathematics book, can I use the related articles of mathematics for the authorship of the book with your permission, the authors? Germany Poul Ah (talk) 13:06, 16 June 2023 (UTC)

As long as you comply with the license. Details and instructions are at Wikipedia:Reusing Wikipedia content. MrOllie (talk) 13:28, 16 June 2023 (UTC)