Talk:Law of cosines (spherical)

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[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

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