Editing Spherical trigonometry (section)

==Cosine rules and sine rules==

===Cosine rules===
{{main|Spherical law of cosines}}

The cosine rule is the fundamental identity of spherical trigonometry: all other identities, including the sine rule, may be derived from the cosine rule:
<math display=block>\begin{align}
  \cos a &= \cos b \cos c + \sin b \sin c \cos A, \\[2pt]
  \cos b &= \cos c \cos a + \sin c \sin a \cos B, \\[2pt]
  \cos c &= \cos a \cos b + \sin a \sin b \cos C.
\end{align}</math>

These identities generalize the cosine rule of plane [[trigonometry]], to which they are asymptotically equivalent
in the limit of small interior angles. (On the unit sphere, if <math>a, b, c \rightarrow 0</math>  set <math> \sin a \approx a </math> and <math> \cos a \approx 1 - \frac{a^2}{2} </math> etc.; see [[Spherical law of cosines]].)

===Sine rules===
{{main|Spherical law of sines}}

The spherical [[Law of sines#Curvature|law of sines]] is given by the formula
<math display=block>\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}.</math>
These identities approximate the sine rule of plane [[trigonometry]] when the sides are much smaller than the radius of the sphere.

===Derivation of the cosine rule ===
{{main|Spherical law of cosines}}

[[File:Spherical trigonometry vectors.svg|thumb|right|200px]]
The spherical cosine formulae were originally proved by elementary geometry and the planar cosine rule (Todhunter,<ref name=todhunter/> Art.37). He also gives a derivation using simple coordinate geometry and the planar cosine rule (Art.60). The approach outlined here uses simpler [[Euclidean vector|vector]] methods. (These methods are also discussed at [[Spherical law of cosines]].)

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Consider three unit vectors {{math|''{{vec|OA}}'', ''{{vec|OB}}'', ''{{vec|OC}}''}} drawn from the origin to the vertices of the triangle (on the unit sphere). The arc {{mvar|{{overarc|BC}}}} subtends an angle of magnitude {{mvar|a}} at the centre and therefore {{math|1={{vec|''OB''}} · ''{{vec|OC}}'' = cos ''a''}}. Introduce a Cartesian basis with {{mvar|{{vec|OA}}}} along the {{mvar|z}}-axis and {{mvar|{{vec|OB}}}} in the {{mvar|xz}}-plane making an angle {{mvar|c}} with the {{mvar|z}}-axis. The vector {{mvar|{{vec|OC}}}} projects to {{mvar|ON}} in the {{mvar|xy}}-plane and the angle between {{mvar|ON}} and the {{mvar|x}}-axis is {{mvar|A}}. Therefore, the three vectors have components:

<math display=block>\begin{align}
  \vec{OA}: &\quad (0,\,0,\,1) \\
  \vec{OB}: &\quad (\sin c,\,0,\,\cos c) \\
  \vec{OC}: &\quad (\sin b\cos A,\,\sin b\sin A,\,\cos b).
\end{align}</math>

The scalar product {{mvar|{{vec|OB}} · {{vec|OC}}}} in terms of the components is
<math display=block>\vec{OB} \cdot \vec{OC} =\sin c \sin b \cos A  +  \cos c \cos b.</math>
Equating the two expressions for the scalar product gives
<math display=block>\cos a  =  \cos b \cos c  +  \sin b \sin c \cos A.</math>
This equation can be re-arranged to give explicit expressions for the angle in terms of the sides:
<math display=block>\cos A = \frac{\cos a-\cos b\cos c}{\sin b \sin c}.</math>

The other cosine rules are obtained by cyclic permutations.

===Derivation of the sine rule ===
{{main|Spherical law of sines}}

This derivation is given in Todhunter,<ref name=todhunter/> (Art.40). From the identity <math>\sin^2 A=1-\cos^2 A</math> and the explicit expression for {{math|cos ''A''}} given immediately above
<math display=block>
\begin{align}
   \sin^2 A &= 1 - \left(\frac{\cos a - \cos b \cos c}{\sin b \sin c}\right)^2 \\[5pt]
   &= \frac{(1-\cos^2 b)(1-\cos^2 c)-(\cos a  - \cos b\cos c)^2}{\sin^2\!b \,\sin^2\!c} \\[5pt]
 \frac{\sin A}{\sin a} &= \frac{\sqrt{1-\cos^2\!a-\cos^2\!b-\cos^2\!c + 2\cos a\cos b\cos c}}{\sin a\sin b\sin c}.
\end{align}</math>
Since the right hand side is invariant under a cyclic permutation of {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} the spherical sine rule follows immediately.

===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.

=== Differential variations ===
When any three of the differentials ''da'', ''db'', ''dc'', ''dA'', ''dB'', ''dC'' are known, the following equations, which are found by differentiating the cosine rule and using the sine rule, can be used to calculate the other three by elimination:<ref>{{cite book |title=A Treatise on Plane and Spherical Trigonometry |author1=William Chauvenet |edition=9th |publisher=J.B. Lippincott Company |year=1887 |isbn=978-3-382-17783-6 |page=240 |url=https://books.google.com/books?id=nWu4EAAAQBAJ}}</ref>

<math display=block>\begin{align}
 da = \cos C \ db + \cos B \ dc + \sin b \ \sin C \ dA, \\
 db = \cos A \ dc + \cos C \ da + \sin c \ \sin A \ dB, \\
 dc = \cos B \ da + \cos A \ db + \sin a \ \sin B \ dC. \\
\end{align}</math>