Trigonometric constants expressed in real radicals in a regular polygon: a vertex, an edge center containing that vertex, and the polygon center. An "n"-gon can be divided into 2"n" right triangles with angles of , 90 − , 90} degrees, for "n" in 3, 4, 5, … Constructibility of 3, 4, 5, and 15-sided polygons are the basis, and angle bisectors allow multiples of two to also be derived. In degree format, sin and cos of 0, 30, 45, 60, and 90 can be calculated from their right angled triangles, using the Pythagorean theorem. In radian format, sin and cos of / 2 can be expressed
Schwarz triangle function general method which works for all right angled triangles for which the two other angles are less than or equal to /4. The previous method for constructing "P", "P", ... is modified by adding an extra triangle each time an angle 3/2 arises at a vertex. The same reasoning applies to prove there is no overlapping and that the tiling covers the hyperbolic upper half plane. On the other hand, the given configuration gives rise to an arithmetic triangle group. These were first studied in . and have given rise to an extensive literature. In 1977 Takeuchi obtained a complete