Special right triangle triangles are used to aid in calculating common trigonometric functions, as below: The 45°–45°–90° triangle, the 30°–60°–90° triangle, and the equilateral/equiangular (60°–60°–60°) triangle are the three Möbius triangles in the plane, meaning that they tessellate the plane via reflections in their sides; see Triangle group. In plane geometry, constructing the diagonal of a square results in a triangle whose three angles are in the ratio 1 : 1 : 2, adding up to 180° or radians. Hence, the angles respectively measure 45° (), 45° (), and 90° (). The sides in this triangle are in the ratio 1 : 1
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