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Spherical trigonometry
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===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]].) <!-- ===================================== I have restored my direct construction for the simple inline maths in this section. Wiki maths markup is still very crude and fails to address the mismatch between text and maths fonts. The result can be very ugly. One day this may improve. ===================================== --> 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''}} Β· ''{{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}} Β· {{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.
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