Editing Spherical trigonometry (section)

===Alternative derivations===
There are many ways of deriving the fundamental cosine and sine rules and the other rules developed in the following sections. For example, Todhunter<ref name=todhunter/> gives two proofs of the cosine rule (Articles 37 and 60) and two proofs of the sine rule (Articles 40 and 42). The page on [[Spherical law of cosines]] gives four different proofs of the cosine rule. Text books on geodesy<ref>{{cite book|year=1880|last=Clarke|first=Alexander Ross|title=Geodesy|url=https://archive.org/details/in.ernet.dli.2015.42772|publisher=Clarendon Press|location=Oxford|oclc=2484948|via=the [[Internet Archive]]}}</ref> and spherical astronomy<ref>{{cite book |year=1977|last=Smart|first=W.M.|title=Text-Book on Spherical Astronomy|publisher=Cambridge University Press|edition=6th|url=https://archive.org/details/textbookonspheri0000smar|at=Chapter 1|via=the [[Internet Archive]]}}</ref> give different proofs and the online resources of [[MathWorld]] provide yet more.<ref>{{MathWorld|title=Spherical Trigonometry|id=SphericalTrigonometry|access-date=8 April 2018}}</ref> There are even more exotic derivations, such as that of Banerjee<ref name="banerjee">{{Citation | last = Banerjee | first = Sudipto | date = 2004 | title = Revisiting Spherical Trigonometry with Orthogonal Projectors | journal = The College Mathematics Journal | volume = 35 | issue = 5 | pages = 375–381 | publisher = Mathematical Association of America | url = https://www.researchgate.net/publication/228849546 | doi = 10.1080/07468342.2004.11922099 | jstor = 4146847 | s2cid = 122277398 | access-date = 2016-01-10 | archive-date = 2020-07-22 | archive-url = https://web.archive.org/web/20200722071405/https://www.researchgate.net/publication/228849546_Revisiting_Spherical_Trigonometry_with_Orthogonal_Projectors | url-status = live }}</ref> who derives the formulae using the linear algebra of projection matrices and also quotes methods in [[differential geometry]] and the group theory of rotations.

The derivation of the cosine rule presented above has the merits of simplicity and directness and the derivation of the sine rule emphasises the fact that no separate proof is required other than the cosine rule. However, the above geometry may be used to give an independent proof of the sine rule.  The [[scalar triple product]], {{math|''{{vec|OA}}'' · (''{{vec|OB}}'' × ''{{vec|OC}}'')}} evaluates to {{math|sin ''b'' sin ''c'' sin ''A''}} in the basis shown. Similarly, in a basis oriented with  the {{mvar|z}}-axis along {{mvar|{{vec|OB}}}}, the triple product {{math|''{{vec|OB}}'' · (''{{vec|OC}}'' × ''{{vec|OA}}'')}}, evaluates to {{math|sin ''c'' sin ''a'' sin ''B''}}. Therefore, the invariance of the triple product under cyclic permutations gives {{math|1=sin ''b'' sin ''A'' = sin ''a'' sin ''B''}} which is the first of the sine rules. See curved variations of the [[law of sines]] to see details of this derivation.