Geometric Interpretation

 

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

Also, we have a straight line of angle θ in respect to the x axis, which begins from the origins of the Cartesian system and intersects the hyperbola at the point A.

Figure 1: Cartesian hyperbola

 Hyperbolic sine (sinhθ) is the length of the y axis, from the origins of the Cartesian system, up to the point of the projection of the point A on the y axis.

Hyperbolic cosine (coshθ) is the length of the x axis, from the origins of the Cartesian system, up to the point of the projection of the point A on the x axis.

Hyperbolic tangent (tanhθ) is the length of the y axis from the origins of the Cartesian system, up to the point of the projection of the line on the y axis, at x=1.