Coxeter decompositions of hyperbolic polygons the sum of the angles of a hyperbolic triangle is less than 180 degrees. Coxeter decompositions are named after Harold Scott MacDonald Coxeter, an accomplished 20th century geometer. He introduced the Coxeter group, an abstract group generated by reflections. These groups have many uses, including producing the rotations of Platonic solids and tessellating the plane. Given a polygon "P", a group "G" can be generated by reflecting "P" around its sides. If the angles of "P" are /"k" for natural numbers "k", then "G" will be discrete. A Coxeter decomposition of a polygon is a decomposition into a finite number
Proofs of trigonometric identities division identity They remain valid for angles greater than 90° and for negative angles. Or Two angles whose sum is π/2 radians (90 degrees) are "complementary". In the diagram, the angles at vertices A and B are complementary, so we can exchange a and b, and change θ to π/2 − θ, obtaining: Identity 1: The following two results follow from this and the ratio identities. To obtain the first, divide both sides of formula_21 by formula_23; for the second, divide by formula_24. Similarly Identity 2: The following accounts for all three reciprocal functions. Proof 2: Refer to the triangle