The generalized law of sines applies to a simplex in space of any dimension with constant Gaussian curvature. Let us work up to that. Initially in two-dimensional space, we define a generalized sine function for a one-dimensional simplex (line segment) with content (length) S in space of constant Gaussian curvature K as gsin S = S - (K S^3)/(3!) + (K^2 S^5)/(5!) - (K^3 S^7)/(7!) + (K^4 S^9)/(9!) - (K^5 S^11)/(11!) + ....
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