Talk:Law of cosines

Correction for solving angle-side-side triangles

In the section where you use the quadratic equation to make a form of the law of cosines that solves angle-side-side triangles, there is an error on the bounds checking. You say that if c>=b, there are no solutions. However:

- If c=b, there is one solution. It is an isosceles triangle with b=c, B=C. The triangle itself is twice the size as the right triangle formed by c=b*sin(C) & B = 90. B and C are acute.

- If c>b, there is also one solution. It is a scalene triangle with B and C being acute. — Preceding unsigned comment added by 69.159.12.95 (talk) 01:35, 26 January 2020 (UTC)

For future reference, WP:SOFIXIT applies. –Deacon Vorbis (carbon • videos) 01:51, 26 January 2020 (UTC)

Another small addition on the angle-side-side solution

Right now it says, "only one positive solution if c = b sin γ". It should say "only one positive solution if c = b sin γ OR if c>=b" — Preceding unsigned comment added by 69.159.12.95 (talk) 06:06, 26 January 2020 (UTC)

WP:SOFIXIT. Also, please place new comments at the end and sign your posts. See Help:Talk for more info. –Deacon Vorbis (carbon • videos) 12:44, 26 January 2020 (UTC)

Demonstration by Chasles relation and scalar product

Will it make sense to add a demonstration with Chasles relation and scalar product? Even if historically it was after, it is a simple and powerful tool to demonstrate it. — Preceding unsigned comment added by 103.139.171.78 (talk) 17:49, 15 July 2020 (UTC)

Proof by application of dot product

Could there be a valid proof using the definition of a dot product. With triangle having side C as the vector A-B, where A,B are the vectors of the triangle legs: then length |C|^2 could be written as |A|^2+|B|^2-2A(dot)B. A(dot)B is defined as |A||B|cos(theta) giving the law of cosines formula. If other editors think this is valid, I could add a formatted proof. Nikolaih^☎️📖 21:40, 6 August 2020 (UTC)

Absolutely. It should be added --83.56.100.228 (talk) 18:37, 10 October 2020 (UTC)

Of course then you have to show why A dot B = |A| |B| cos (theta) is a reasonable definition, since usually dot products are defined by coordinates. Dot product#Scalar projection and first properties Wqwt (talk) 17:44, 15 June 2024 (UTC)

Requires proof reading?

Perhaps i'm mistaken but angle ABC in Fig 2 appears acute rather than obtuse to me. Proof reading anyone? — Preceding unsigned comment added by 198.24.255.98 (talk) 16:24, 22 September 2020 (UTC)

No, it's definitely obtuse (unless there's some issue with it being displayed in a weird aspect ratio). –Deacon Vorbis (carbon • videos) 16:55, 22 September 2020 (UTC)

The angle ABC in Fig 2 clearly looks acute on my monitor. Just to be clear an acute angle is smaller than 90 degrees and an obtuse angle is greater than 90 degrees but less than 180 degrees. It makes the whole section unreadable, and brings into question the correctness of articles on the site. — Preceding unsigned comment added by 2605:A601:A706:8700:F91E:CE18:C1EC:7244 (talk) 23:19, 27 September 2020 (UTC)

Oh, I think I see the confusion. Angle ABC is acute, but triangle ABC is obtuse, because angle ACB is an obtuse angle. –Deacon Vorbis (carbon • videos) 23:28, 27 September 2020 (UTC)

Right, ABC is an angle not a triangle. 'Triangle ABC' is confusing if not complete nonsense. — Preceding unsigned comment added by 198.24.255.98 (talk) 23:35, 2 October 2020 (UTC)

It's perfectly clear what $\Delta ABC$ means: the triangle defined by the vertices A, B, C. Wqwt (talk) 17:45, 15 June 2024 (UTC)

Recent reverts by Jacobolus

There we go, the sentence you removed twice without explaining your rationale here is explicitly stated in the cited source, I quote : "IN the fifteeth century, the Persian astronomer and mathematician al-Kashi provided accurate trigonometric tables and expressed the theorem in a form suitable for modern usage." since apparently, your concerns are about the traduction of the French word "Théorème" in English, I will reword that. Best.---Wikaviani ^{(talk) (contribs)} 08:43, 29 January 2025 (UTC)

Yes, but the source's claim is (at best) extremely misleading and confusing, so it's a poor idea to uncritically repeat it. Personally I'd just as soon entirely remove this bit of trivia about the name a small minority of French sources use. But I also don't mind keeping the claim as long as it remains as a cute one-sentence aside without making inaccurate statements. We are using Pickover only as a source for the claim that some French sources use this name, not for any less credible statements.

To elaborate: "modern usage" is completely undefined, and al-Kashi did not present the "law of cosines" as anything like the algebraic expression found in modern textbooks; presumably by "modern usage" Pickover means "finding the third side of a triangle given the other two sides and their included angle" but if so we should say that directly instead of something totally vague.... but we shouldn't say that al-Kashi was the first to this idea because he wasn't: people had been solving such triangles for centuries before, and even if we restrict ourselves to al-Kashi's specific trigonometric method, it is essentially identical to al-Tusi's from 2 centuries before, except with all of the steps included in one place (al-Tusi reduces the problem to the right-triangle case which he had previously presented). Please do not add this passage back to the article, it is not appropriate for inclusion. –jacobolus (t) 10:04, 29 January 2025 (UTC)

Aside: I'm surprised you are so keen on using Clifford A. Pickover as a source, considering how much you belabored weird criticisms of authors who you considered to be insufficiently expert over at Talk:Binomial theorem § History section last month. Pickover is an amateur who writes pop science books, not any kind of historian. –jacobolus (t) 10:22, 29 January 2025 (UTC)

Ok, so now, you say that there is a need for historians here while you blatantly said the opposite few weeks ago, ok, no problem, i will add some other sources. however, all this sounds like WP:JUSTDONTLIKEIT and WP:POV.---Wikaviani ^{(talk) (contribs)} 18:57, 29 January 2025 (UTC)

No, I did not say that. I just don't think your previous standard was ever real, but seems just like an excuse for pushing some kind of ideological agenda, as evidenced by the speed and thoroughness with which you ignore your previously religiously argued standard whenever convenient. –jacobolus (t) 19:01, 29 January 2025 (UTC)

You seem to ignore yours too, anyway, other sources have been added, they roughly say the same thing that is said by Pickover. BTW, I have not any agenda here and I never said or implied that our readers are stupid, keep focused on content, do not comment on editors. You can consider this post of mines as a last warning.---Wikaviani ^{(talk) (contribs)} 19:21, 29 January 2025 (UTC)

A warning of what? Please see WP:BRD. After being reverted, you should try to build consensus for your preference on the discussion page, rather than edit warring to install your preferred language. The text you want to add remains unacceptably problematic. Do not add it again without consensus. –jacobolus (t) 23:08, 29 January 2025 (UTC)

A warning for edit warring and making comments about users. Of course, al-Kashi did not give the law of cosines in its modern form, i.e. with modern algebric notations, that were introduced from Viète and his successors, but al-Kashi is the first to have given, in his work "Miftah al-hisab" the formula for a random triangle. If the phrasing is not convenient for you, then let's find another one, but edit-warring you do is not a solution. First, you discussed my first edit with 3 concerns, I fixed two of them and the third one was not clear, I didn't know if you were talking about my explanation or the source's explanation, next you discussed the quality of the source (Pickover) and when I added other sources, you said that "the source is not the problem". Thus, I must confess that I'm a bit baffled, what is the problem exactly ?---Wikaviani ^{(talk) (contribs)} 09:59, 30 January 2025 (UTC)

I took the trouble of tracking down al-Kashi's text (in translation) making a picture to go along with it, and then describing the relation between his text and the same statement in modern algebraic notation. As a result, our article here does a better job describing what al-Kashi did and how it relates to earlier examples than any other sources I know of. We don't need to add promotional text puffing up al-Kashi beyond that, overstating how often his name is attached to this in France, etc. The description as it is in the article speaks for itself.

What this section of the article does need, if anyone feels motivated, is a better description of the later history, from the past few centuries. –jacobolus (t) 15:49, 30 January 2025 (UTC)

I disagree, we are are editing an online encyclopedia, there is no puffery in presenting fairly the contribution of the mathematicians who studied the law of cosines. As Youcef Guergour says in his paper, "This proposition has been proved in a number of ways. Despite Euclid's informations on the case of any triangle in different parts of the elements, mathematicians have only studied the case of right-angled triangles. In the 15th century, the mathematician from Samarkand, al-Kashi, in his book The Keys of Arithmetic, described the general case of any triangle by introducing trigonometry. In the 16th century, François Viète rediscovered a demonstration similar to that of al-Kashi."

This is a key development in the history of this law, namely, the introduction of trigonometry (that did not exist at the time of Euclid) and the statement of the law in the general case. this is not "puffery" to introduce something about this.---Wikaviani ^{(talk) (contribs)} 17:34, 30 January 2025 (UTC)

The claim that Al-Kashi's method of solving triangles (or variant of the 'law of cosines' or whatever) is the first "in a form suitable for modern usage" is not supportable to write in Wikipedia's authorial voice. At best it's misleading, but I would call it outright inaccurate. There's really nothing about earlier expressions of the same idea that was "unsuitable" for solving triangles, and al-Kashi's version is also not really "modern", if by that we mean something that would show up in a modern textbook. In theory Wikipedia could supportably make a claim along the lines of "according to pop science author Clifford Pickover this was the first version of the law of cosines in a form suitable for modern use", but that statement is frankly not useful to readers. Readers benefit much more from us explicitly describing what al-Kashi did and how it related to earlier and later expressions than from us making vague and subjective claims about who deserves credit. Readers are not stupid: they can draw their own conclusions about the relevance if we just stick to the historical record instead of trying to force personal subjective conclusions onto them. Take a look at MOS:PUFFERY and WP:SOAPBOX for more on this point. –jacobolus (t) 19:08, 29 January 2025 (UTC)

The phrasing in a form suitable for modern usage conveys no meaningful information whatsoever. XOR'easter (talk) 01:49, 30 January 2025 (UTC)

Ok, if that phrasing is not convenient, let's find another one.---Wikaviani ^{(talk) (contribs)} 10:00, 30 January 2025 (UTC)