the stereographic projection of the point onto the -axis with the center of projection at . Thus, the tangent half-angle formulae give conversions between the stereographic coordinate on the unit circle and the standard angular coordinate . Then we have and By eliminating phi between the directly above and the initial definition of , one arrives at the following useful relationship for the arctangent in terms of the natural logarithm In calculus, the Weierstrass substitution is used to find antiderivatives of rational functions of and . After setting This implies that for some integer , and therefore Hyperbolic identities One can play an entirely analogous game with the hyperbolic functions. A point on (the right branch of) a hyperbola is given by . Projecting this onto -axis from the center gives the following: with the identities and Finding in terms of leads to following relationship between the inverse hyperbolic tangent and the natural logarithm: The Gudermannian function Comparing the hyperbolic identities to the circular ones, one notices that they involve the same functions of , just permuted. If we identify the parameter in both cases we arrive at a relationship between the circular functions and the hyperbolic ones. That is, if then where is the Gudermannian function. The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers. The above descriptions of the tangent half-angle formulae (projection the unit circle and standard hyperbola onto the -axis) give a geometric interpretation of this function. Rational values and Pythagorean triples If is a rational number then each of , , , , , and will be a rational number (or be infinite). Likewise if is a rational number then each of , , , , , and will be a rational number (or be infinite). This is helpful with Pythagorean triples; each interior angle has a rational sine because of the SAS area formula for a triangle and has a rational cosine because of the Law of Cosines. These imply that the half-angle tangent is necessarily rational. Vice versa, when a half-angle tangent is a rational number in the interval then the full-angle sine and cosine will both be rational, and there is a right triangle that has the full angle and that has side lengths that are a Pythagorean triple. Generally, if is a subfield of the complex numbers then implies