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Exact algebraic expressions for trigonometric values are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. All values of the sines, cosines, and tangents of angles at 3° increments are derivable in radicals using identities—the half-angle identity, the double-angle identity, and the angle addition/subtraction identity—and using values for 0°, 30°, 36°, and 45°. Note that 1° = π/180 radians . According to Niven's theorem , the only rational values of the sine function for which the argument is a rational number of degrees are 0, 1/2, and 1. [ edit ] Fermat number The list in this article is incomplete in at least two senses.
Exact trigonometric constants
Wythoff construction
Wythoffian constructions from 3 mirrors forming a right triangle. In geometry , a Wythoff construction , named after mathematician Willem Abraham Wythoff , is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. [ edit ] Construction processCoxeter–Dynkin diagram
Coxeter–Dynkin diagrams for the fundamental finite Coxeter groups Coxeter–Dynkin diagrams for the fundamental affine Coxeter groups In geometry , a Coxeter–Dynkin diagram (or Coxeter diagram , Coxeter graph ) is a graph with numerically labeled edges (called branches ) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes ). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet ) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge ).