Hyperbolic Trigonometric Functions

the Exponential Function, a review

Periodicity is a characteristic feature of Elliptic Functions. Elliptic Functions may be either doubly-periodic, singly-periodic, or (triviallly) non-periodic. The Exponential Function is the singly-periodic Elliptic Function. Thus, the Exponential Function possesses the modulus of periodicity of 2 pi i

exp(x + 2 pi i) = exp(x) for any x in Complex.

And, by saying that 2 pi i is the "modulus of periodicity" we mean that 2 pi i is the smallest, in magnitude, constant for which the foregoing identity is true. Incidentally, we thus have defined pi. An approximate value of pi is 3.14159, a Real number. Each of the trigonometric funcions will inherit this modulus of periodicity, except that the tangent and cotanget functions have a modulus of periodicity only half the size.

Possesion of an addition theorem is another characteristic feature of Elliptic Functions. The addition theorem for the Exponential Function is

exp(x + y) = exp(x) exp(y) for any (x, y) in the Cartesian product of Complex by Complex.

The third characteristic property of the Elliptic Functions is that they satisfy a certain differential equation. Thus, the exponential function may be characterized by the differential equation

d exp(x) / dx - exp(x) = 0, with exp(0) = 1,

for any x in Complex.

The McLaurin's series for the exponential function is

exp(x) = 1 + x + x^2 / 2 + x^3 / 3 + ... + x^n / n! + ....

Hyperbolic Trigonometric Functions


For an x in Complex, we define the nine Hyperbolic Trigonometric functins as follows:


For any x in the Cartesian product of Complex by Complex, from the foregoing, the following identities are obvious:

Addition Theorems


For any (x, y) in the Cartesian product of Complex by Complex, from the definitions of the Hyperbolic Trigonometric function, the following real addition theorems are obvious:


By substitution of the addition formulae into the right-hand side of the following, one may obtain these real product theorems:

Sums or Differences

Let u = x + y and v = x - y. Substitution in the foregoing three equations yields the Sums or Differencess


For any (x, y) in the Cartesian product of Complex by Complex, from the foregoing read addifion theorems, with the replacement of y by i y and the use of the circular trigonometric functions, the following complex addition theorems are obvious

The special cases where x is zero are as follows

On the other hand, we may invert the first three of the foregoing complex addition theorems. Set the right-hand side equal to u + i v. Then collect the real and imaginary parts on one side of the equation; each part has to be zero. Then by employing the identities, we obtain

Double-Angle Formulae

In the foregoing real addition theorems, take y equal to x, to yield the following double-angle formulae:

By the use of the first of the identities, the double angle formula for the hyperbolilc cosine may be written in either of the additinal two forms:

Half-Angle Formulae

The half-angle formulae are as follows:

Inverse Hyperbolic in Terms of Logarithms

Let y = Arcsinh(x) and solve for x to obtain

x = sinh(y) = (exp(y) - exp(- y)) / 2 = (exp(2 y) - 1) / exp(y).

Solution of this quadratic equation yields the forst of the following formulae of the inverse hyperboic trigonometric functions in terms of the logarithmic function.


A parametric equation of a hyperbola, in the Cartesian product of Complex by Complex, is given by

(x, y) = (a cosh(t), b sinh(t)) for any t in Complex and any constant (a, b), called the semi-axes, in the Cartesian product of Complex by Complex. In the case of real t, this formula yields only one branch of the hyperbola. To obtain the other branch, change the sign of a.

Passive (that is with time being unknown) navigation employs hyperbolas and hyperboloids of revolution of two-sheets.

Historically, these functions have been called hyperbolic because of this parametrization of a nyperbola.



These derivative formulae are obvious from the definition of the functions. For any x in Complex we have the following derivative formulae:



Let x = sinh(y) and differentiate it to obtain dx / dy = cosh(y). Employ the appropriateidentities to obtain dx / dy = sqrt(1 + (sinh(y))^2). Then dy / dx = 1 / sqrt( + (sihn(y))^2). Thus, we have obtained the first of the derivative formulae of the inverse hyperbolic trigonometric functions

Their primary utility is as antiderivatives.


The Riemann Integral is an ant-derivative, thus each of these integral formulae may be verified by differentiation. For any x in Complex we have the integral formuale:

Infinite Expansions

McLaurin's Series

The McLaurin's series for the hyperbolic sine and cosine may be obtained from their definitions and the series for the exponential function.

Take the infinite Geometric series

1 / (1 - x) = 1 + x + x^2 + x^3 + ... + x^n + ....

for any x, in Complex, whose the absolute value is less than one. Replace x by x^2 and integrate to obtain the McLaurin's series for the hyperbolic arctangent.

Infinite Products

The hyperbolic infinite products are not interesting. They have to be obtained from the corresponding circular infinite products, employing the purely imaginary relationship between the corresponding functions.

Circular Trigonometric Functions

The purely imaginary counterpart of the Circular Trigonometric functions is called the Circular Trigonometric functions.

Copywrite 1997 R. I. 'Scibor-Marchocki last modified on Friday 15-th of August 1997.