Trigonometric functions are intimately related to triangle geometry. Functions like sine and cosine are often introduced as edge lengths of right‐angled triangles. Hyperbolic functions occur in the theory of triangles in hyperbolic spaces.
Lobachevsky (1829) and J. Bolyai (1832) independently recognized that Euclid's fifth postulate—saying that for a given line and a point not on the line, there is exactly one line parallel to the first—might be changed and still be a consistent geometry. In the hyperbolic geometry it is allowable for more than one line to be parallel to the first (meaning that the parallel lines will never meet the first, however far they are extended). Translated into triangles, this means that the sum of the three angles is always less than .
A particularly nice representation of the hyperbolic geometry can be realized in the unit disk of complex numbers (the Poincaré disk model). In this model, points are complex numbers in the unit disk, and the lines are either arcs of circles that meet the boundary of the unit circle orthogonal or diameters of the unit circle.
The distance between two points (meaning complex numbers) and in the Poincaré disk is:
The attractive feature of the Poincaré disk model is that the hyperbolic angles agree with the Euclidean angles. Formally, the angle at a point of two hyperbolic lines and is described by the formula:
In the following, the values of the three angles of an hyperbolic triangle at the vertices , , and are denoted through , , and . The hyperbolic length of the three edges opposite to the angles are denoted , , and .
The cosine rule and the second cosine rule for hyperbolic triangles are:
The sine rule for hyperbolic triangles is:
For a right‐angle triangle, the hyperbolic version of the Pythagorean theorem follows from the preceding formulas (the right angle is taken at vertex ):
Using the series expansion at small scales the hyperbolic geometry is approximated by the familar Euclidean geometry. The cosine formulas and the sine formulas for hyperbolic triangles with a right angle at vertex become:
The inscribed circle has the radius:
The circumscribed circle has the radius:
As rational functions of the exponential function, the hyperbolic functions appear virtually everywhere in quantitative sciences. It is impossible to list their numerous applications in teaching, science, engineering, and art.
