3rd Year Honours Mathematical Physics
Special Relativity
Hyperbolic Trigonometric Functions

Brian Dolan

You are familiar with ordinary trigonometric functions and their relation to complex numbers. A complex number z=x+iy can be represented by a point in a two dimensional plane (an Argand diagram) with Cartesion co-ordinates (x,y). Using polar co-ordinates, $x=r\cos\varphi$, $y=r\sin\varphi$ one can write $z=r(\cos\varphi +i\sin\varphi)=r\hbox{e}^{i\varphi}$. This relies on the fact that the complex exponential can be written as

$$\hbox{e}^{i\varphi}=\cos\varphi +i\sin\varphi.$$ (1)

Using this expression, we can write $\cos\varphi$ and $\sin\varphi$in terms of the exponential

$$\cos\varphi={1\over 2}\left(\hbox{e}^{i\varphi}+\hbox{e}^{-i\varphi}\right)$$ (2)

$$\sin\varphi={1\over 2i}\left(\hbox{e}^{i\varphi}-\hbox{e}^{-i\varphi}\right).$$ (3)

It is easy to verify the trigonometric relation $\cos^2\varphi+\sin^2\varphi=1$from these formulae. This leads to the algebraic expression for the equation for a circle of radius r in the plane,

$$x^2+y^2=r^2\bigl(\cos^2\varphi+\sin^2\varphi\bigr)=r^2,$$ (4)

and the angle $\varphi$ parameterises points on the circle, $0\le \varphi<2\pi$ . The exponential form is well suited to proving the usual trigonometric relations,

$$\cos(\varphi \pm \varphi^\prime)=\cos\varphi\cos\varphi^\prime \mp \sin\varphi\sin\varphi^\prime$$ (5)

$$\sin(\varphi \pm \varphi^\prime)=\sin\varphi\cos\varphi^\prime \pm \cos\varphi\sin\varphi^\prime,$$ (6)

from which the formulae

$$\tan(\varphi \pm \varphi^\prime)=\tan\varphi+\tan\varphi^\prime \over 1 \mp \tan\varphi\tan\varphi^\prime$$ (7)

follows.

The behaviour of sin and cos under differentiation can also be determined easily from the exponential representation. Since ${d \hbox{e}^{i\varphi}\over d\varphi}=i\hbox{e}^{i\varphi}$ one finds ${d \sin\varphi\over d\varphi}=\cos\varphi$ and ${d \cos\varphi\over d\varphi}=-\sin\varphi$ by taking real and imaginary parts.

Hyperbolic trigonometric functions are related to hyperbolae in a manner similar to that in which trigonometric functions are related to circles. They are actually much simpler, because they do not require complex exponentials. Define the hyperbolic cosine, cosh, and the hyperbolic sine, sinh, in the following manner,

$$\cosh\alpha={1\over 2}\bigl(\hbox{e}^\alpha + \hbox{e}^{-\alp... ...pha={1\over 2}\bigl(\hbox{e}^\alpha - \hbox{e}^{-\alpha}\bigl),$$ (8)

where $\alpha$ is a real parameter.

The hyperbolic cosine, cosh, has the value 1 at $\alpha=0$and is an even function of $\alpha$, $\cosh(\alpha)=\cosh(-\alpha)$, while the hyperbolic sine, sinh, has the value 0 at $\alpha=0$and is an odd function of $\alpha$, $\sinh(\alpha)=-\sinh(-\alpha)$.

Here are graphs of these hyperbolic functions:

The hyperbolic cosine function, cosh.

The hyperbolic sine function, sinh.

It follows from the definitions given above that

$$\cosh^2\alpha - \sinh^2\alpha = 1.$$ (9)

In analogy with the usual trigonometric functions, points in the (x,y) plane can be parameterised by two new variables, $\rho$ and $\alpha$, defined in terms of x and y by

$$x=\rho\cosh\alpha\qquad\hbox{and}\qquad y=\rho\sinh\alpha.$$ (10)

Then curves of constant $\rho$ are described by the algebraic equation

$$x^2 - y^2 = \rho^2.$$ (11)

These curves are hyperbolae in the (x,y)-plane, with points on the hyperbolae parameterised by the variable $\alpha$.

The above definitions lead immediately to hyperbolic analogues of the formulae for addition of arguements

$$\cosh(\alpha\pm \alpha^\prime)=\cosh\alpha\cosh\alpha^\prime \pm \sinh\alpha\sinh\alpha^\prime,$$ (12)

$$\sinh(\alpha \pm \alpha^\prime)=\sinh\alpha\cos\alpha^\prime \pm \cos\alpha\sin\alpha^\prime$$ (13)

$$\tanh(\alpha \pm \alpha^\prime)={\tanh\alpha\pm\tanh\alpha^\prime \over 1 \pm \cosh\alpha\sinh\alpha^\prime},$$ (14)

where $\tanh\alpha={\sinh\alpha\over\cosh\alpha}$.

A graph of the hyperbolic tangent function, tanh, looks like this:

The derivative formulae,

$${d \sinh\alpha\over d\alpha}=\cosh\alpha$$ (15)

$${d \cosh\alpha\over d\alpha}=\sinh\alpha$$ (16)

$${d \tanh\alpha\over d\alpha}={1\over\cosh^2\alpha},$$ (17)

also follow from the definition. You are encouraged to derive these formulae, using the definitions of $\sinh\alpha$ and $\cosh\alpha$ given above, in terms of $\hbox{e}^\alpha$.