Talk:Spherical law of cosines

Variant for angles?
An anonymous contributor just added, without sources:


 * The law of cosines could be also used to solve for angles, in this case it states:
 * $$\cos(A) = -\cos(B)\cos(C) + \sin(B)\sin(C)\cos(a) \,$$

I haven't seen this before, and since no sources were provided and I don't have time to check it right now, I've removed it from the article. Feel free to provide a reputable source and add it back. Or maybe there is some trivial argument to derive it from the ordinary form of the law of cosines that I'm missing right now?

—Steven G. Johnson 15:48, 3 July 2006 (UTC)

This formula should be restored. Its proof is very simple: one should take a spherical triangle dual to the given one. For it the following equalities hold $$A' = \pi - a, \quad B' = \pi - b, \quad C' = \pi - c$$ $$a' = \pi - A, \quad b' = \pi - B, \quad c' = \pi - C$$ Taking $$a,b,c,A,B,C $$ from here and dropping primes, one obtains the mentioned result. In the Russian literature this is known as the second spherical cosine theorem, in contrast to the first one.Aburov 22:02, 7 March 2007 (UTC)


 * We still need a reference for this. —Steven G. Johnson (talk) 04:44, 26 August 2008 (UTC)


 * The formula is entirely legit; it's probably in Bowditch (I'll confirm that) and it's certainly in other collections of spherical-trig formulas. Tim Zukas (talk) 16:49, 24 August 2010 (UTC)


 * I already added a reference for this formula some time ago. — Steven G. Johnson (talk) 09:25, 25 August 2010 (UTC)

Mnemonic
"The sea is sissy and crass." 1)   c  =  c  c   +  c s s  2)   cos = cos cos + cos sin sin 3)    a       b     c       A     b     c  4) cos a = cos b cos c + cos A sin b sin c

Q.E.D.

For angles, just change the sign and cases: cos A = cos B cos C - cos a sin B sin C

(Invented by me, Marshall Price of Miami, while sailing to Newport, RI, circa 1983. All rights abandoned.) D021317c 23:42, 23 March 2007 (UTC)

(My username is now "Unfree".) Unfree (talk) 19:16, 31 July 2009 (UTC)

Incidentally, the mnemonic for angles can be thought of as "The sea is sissy, not crass". Unfree (talk) 19:18, 31 July 2009 (UTC)

Unit sphere?
All the "unit sphere" stuff is utterly off-topic. The formula works for _all_ spheres, including the celestial sphere, which theoretically has an infinite radius, or none at all. The same goes for "radians." You can use degrees, grads, radians -- any angular measure your heart desires. The law only involves sines and cosines anyway, and they are dimensionless. It's great for navigation, in which it's the latitude and longitude (angles) that matter, and degrees are always used. If you need to convert to distances, just multiply the degrees by the circumference of the earth and divide by 360 degrees. Occasionally, you might need the law of sines for spherical triangles, but it's so simple it can't be forgotten. It simply says that the ratios of the sides' sines to the angles' sines are all equal. D021317c 00:08, 24 March 2007 (UTC)


 * Well, by immediate inspection the second law can't work for arbitrary radii. It has


 * cos(A) = -cos(B)cos(C) + sin(B)sin(C)cos(a)


 * Imagine inflating the sphere by some amount, but projecting the triangle onto the new sphere. So A, B, and C are exactly as before (since they are the dihedral angles at the origin, and thus don't depend on the radius), but a is defined here as the 'arc-length' which does depend on the radius, which is clearly impossible unless a is zero. So either the article defines 'a' incorrectly, or the law only holds for the unit sphere (incidentally Weisstein's world of math also includes this problem because it explicitly says that a is the arc-length). Idmillington (talk) 18:47, 10 October 2009 (UTC)


 * The usual assumption when reading the formulas of spherical trigonometry is that the sides of a spherical triangle (a, b and c) are measured not by their length but by the angle they subtend at the center of the sphere. If you take another look at the law of cosines for sides (the subject of this article) you'll notice that it too makes no sense if you consider the lengths of the sides to be actual lengths rather than angles. Tim Zukas (talk) 16:47, 24 August 2010 (UTC)

Broken links
Some of the links on the page referred to in "Romuald Ireneus 'Scibor-Marchocki, Spherical trigonometry, Elementary-Geometry Trigonometry web page (1997)" are broken, but by changing the extensions of the URLs from ".htm" to ".html", most of them work.D021317c 00:31, 24 March 2007 (UTC)

Origins of formula
Is there any one person (or group) that originated the spherical law of cosines? —Preceding unsigned comment added by 146.227.1.12 (talk) 12:45, 22 August 2009 (UTC)

Consistency
The first paragraph defines lowercase letters as sides, uppercase letters as angles. Then we present a formula that takes the sine/cosine of both types of variables, only later adding that for the special case of the unit sphere, the lengths of the sides are numerically identical (in radians) to the central angles subtended by those sides.

Given that this is a narrowly focused article written specifically to explain a single formula, this seems sloppy and unnecessarily confusing. I would encourage someone who knows the subject matter to create a better graphic showing and labeling the central angles separately from the sides of the spherical triangle, and listing formulas that use only angles as arguments to trig functions.

Ma-Ma-Max Headroom (talk) 03:06, 11 March 2011 (UTC)


 * I couldn't agree with you more, the notation in this article was driving me nuts with it using lengths as angles in the arguments to functions which require angles. The article is simply wrong in this regard and requires some a/R, b/R, c/R substitutions for a, b, and c, when used in the trig function arguments. MrInteractive (talk) 00:14, 16 August 2022 (UTC)

Eliminating some proofs
We have four proofs in the article. I propose that we remove the first two. I don't find them to be particularly elegant or intuitive. What do you think? 64.132.59.226 (talk) 15:21, 13 February 2018 (UTC)


 * I am going ahead and making this edit. If you disagree, please consider this an instance of the BOLD, revert, discuss cycle; please undo the edit to the extent necessary and comment here as to what you would like to see and why.  64.132.59.226 (talk) 15:11, 20 February 2018 (UTC)