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...Sphericalgeometry let abc beatriangleintheeuclideanplane fromnowon weindicatetheinterior angles a cab b c bca attheverticesmerelyby the sides of length bc and ca then make an angle cosine rule states that abcosc if it reduces to pythagoras theorem is easily proved by constructing say altitude aa h take be base triangle so height draw picture now apply aab aac get bcosc bsinc therulefollowsbyeliminating thesinerule sina sinb sinc can also using since csinb andtherestfollowsbysymmetry important realize sine deduced algebraically from latter tells us cos numerator on right hand side when expanded symmetric in follows we replace or left because theaimofthissectionistoproveanalogousformulasforsphericaltriangles spherical triangles vertices fix cartesian coordinates space with origin o consider sphere s p d x y z centred at word geometry refers exclusively surface not inside position vector v op any point unit i e norm one kvk suppose are three vectors representing points corresponding will ...