2. The hyperbolic functions

Given in the cartesian plane xOy the graph of the equilateral hyperbola δ with center in the origin O and unitary real semiaxis, whose equation is , let V be the point where δ intersects the positive semiaxis of abscissas and P(x;y) any of its points and P' the point of δ symmetrical of P with respect to the x axis.

The coordinates x and y of P are functions of the area OPVP'. We can therefore express x and y as parametrical functions of this area α.

These functions are said respectively hyperbolic cosine and sine of α.

So we have

and, by the equation of δ,

The ratio between hyperbolic sine and cosine is said hyperbolic tangent

By definition we have

The cosine is an even function, while the sine and the tangent are odd functions, therefore the graph of the cosine is symmetrical with respect to the y axis and the graphs of the sine and the tangent are symmetrical with respect to the origin.

The following Javascript application allows you to calculate the values of the three functions.
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The more P approaches V, the more the area OPVP' is indistinguishable from an isosceles triangle having height OV=1 and base PP' = 2 sinh α.

In every triangle the ratio between the double of its area and its base is the height. So we can write

If we move the point P(x;y) around δ increasing the area α so that it becomes α+β, we obtain a new point Q(X;Y) of δ, such that , where

If we apply to the point P(x ; y) of δ the transformation

we just obtain that , so the transformation (2.5) transforms the point P in the point Q.

In general, the transformations like (2.5) transforms an equilateral hyperbola in itself. These transformations are said hyperbolic rotations.

So we have

These (2.6) are the addition formulas for the cosine and the sine.

From these we can easily deduce many other useful identities, like the double-angle formulas

Moreover we can deduce the prosthaphaeresis formulas

The derivative of sinh x with respect to x, by definition, is

From prosthaphaeresis the numerator is

so

The derivative of cosh x with respect to x, by definition, is

From prosthaphaeresis the numerator is

so