Talk:Law of sines

Should it be inscribed angles instead of central angles in the "proof" of "Relation to the circumcircle"? Angles {\displaystyle \angle C}{\displaystyle \angle C} and {\displaystyle \angle D}{\displaystyle \angle D} are inscribed angles that intercept the same arc AB. — Preceding unsigned comment added by 103.139.171.78 (talk) 14:30, 15 July 2020 (UTC)

I tried to determine where Brahmagupta's version of the Law of Sines comes from and followed the reference in this article. However, I found the reference "Wilson, H.J.J., Eastern Science, John Murray Publishers, 1952, p46" is incorrect. As a result, the reference page "Law of sines#cite ref-2" is also incorrect. The author should be H. J. J. Winter rather than H. J. J. Wilson. This is the book on Amazon: https://www.amazon.com/dp/B0000CI5YE/ref=pe_386300_440135490_TE_simp_item_image, which shows clearly that the author is H. J. J. Winter rather than H. J. J. Wilson.

Could someone modify the original page?

Thanks. Ching-Kuang Shene (talk) 09:23, 4 April 2024 (UTC)

The history section here is incomplete and kind of a mess. If I get some time and feel motivated at some point, I might try to find some better sources and somewhat rewrite it. Feel free to give it a shot though.

As for this specific point, the author is HJJ Winter. Here's a scan, https://archive.org/details/easternscienceou0000wint/page/46/ (you need to make a free Internet Archive account and "check out" the book to view this page). –jacobolus (t) 19:02, 4 April 2024 (UTC)

@XOR'easter thanks for the moderate cleanup here. We should probably not directly attribute these uncontroversial claims to specific scholars at all, but just make direct statements giving sources in the footnotes. –jacobolus (t) 22:25, 19 June 2024 (UTC)

Though while we're at it, we might want to be a bit cagier about crediting Brahmagupta; the example involved is an equation that is algebraically related to the law of sines, used for calculating the distance from the Earth to another planet in a planetary theory involving epicycles, but is not quite the same relation, and is not treated as a general tool for solving triangles. Here's what Van Brummelen says: "Whether or not this apparent use of the Law of Sines allows us to claim independent knowledge of it for Brahmagupta is a delicate matter..." It might be best to directly describe what Brahmagupta did, including a diagram, instead of speculating about what he might have thought about it. –jacobolus (t) 23:24, 19 June 2024 (UTC)

What winter is saying is that Brahmagupta stated a rule which we now express in the form 2r=a/sin a = b/sin b = c/sin c. He never said that he stated the given formula. Hu741f4 (talk) 04:56, 30 October 2024 (UTC)

Yes, but that's also not really true, or at least is substantially misleading without a lot of careful qualification and explanation. –jacobolus (t) 05:21, 30 October 2024 (UTC)

Shouldn't we move the spherical law of sines to an article of its own? Hu741f4 (talk) 22:16, 21 July 2025 (UTC)

Yes, this could support its own article. Feel free to write one. However, a significant summary should remain at this page (ideally more info than is here currently, though the proofs can be skipped or shortened). There's not really any value in just moving the current material to a new page unless you intend to spend at least several hours working on it. –jacobolus (t) 22:48, 21 July 2025 (UTC)

As you may know, there is already a Spherical law of cosines page separate from the Law of cosines page. Maybe something about how that is done will be helpful in the present case. —Quantling (talk | contribs) 00:30, 22 July 2025 (UTC)

Unfortunately it's pretty mediocre though. –jacobolus (t) 01:56, 22 July 2025 (UTC)

@Jacobolus https://archive.org/details/historyofmathema00boye/page/158/mode/1up do talks about Euclid as it states that

Propositions II.12 and 13 of the Elements, for example, are the laws of cosines for obtuse and acute angles respectively, stated in geometric rather than trigonometric language and proved by a method similar to that used by Euclid in connection with the Pythagorean theorem. Theorems on the lengths of chords are essentially applications of the modern law of sines. We have seen that Archimedes' theorem on the broken chord can readily be translated into trigonometric language analogous to formulas for sines of sums and differences of angles." Myuoh kaka roi (talk) 21:13, 22 September 2026 (UTC)

The law of cosines is a different identity, not the same as the law of sines. "Essentially applications of the modern law of sines" is not the same thing as a statement of the identity, even in geometric form. Is there a more specific analysis of which "applications" we are talking about? –jacobolus (t) 03:11, 28 September 2026 (UTC)

I see we have sections dedicated to non-zero curvature (where the sin A is divided by the circumference of the circle with radius equal to a; and likewise for (B, b) and (C, c)) and for higher dimensions (where a polar sine is divided by the hypervolume of the opposing facet). Is there any formulation that applies to both curvature and higher dimensions? I am guessing from the complicated Schläfli functions that there is no simple formulation, but if I'm wrong and you know about it, please add something to the article. —Quantling (talk | contribs) 17:05, 14 October 2026 (UTC)

I think trying to write the spherical, flat, and hyperbolic versions as one formula using a wonky curvature-dependent sine function seems like a party trick, and not that informative. Is there a better source than Mathworld? I'd frankly probably recommend just removing that section.

But we should probably discuss the situation for the Lorentzian plane, the de Sitter plane, the "anti-Euclidean plane", etc. (see Yaglom 1968). I've definitely seen sources discussing the law of sines in such general context, though I don't have one immediately at hand. –jacobolus (t) 19:08, 14 October 2026 (UTC)