The trigonometric functions are derived from an orthogonal triangle which is based on the Euclidean geometry. In non Euclidean geometry we may have an orthogonal triangle in which the summation of its angles may not be 180 degrees, it is always less than 180 degrees. This is because its sides can be curved, so the angles are narrowed. In this case we define the hyperbolic trigonometric functions.
Consider the Cartesian system and the hyperbolic function