The Argument to the Hyperbolic Functions

The unit hyperbola is the curve satisfying the equation

(1)    

The hyperbolic cosine and sine are given by:

(2)    

and they satisfy the identity

(3)    

which implies that the point (cosh(u), sinh(u)) lies on the unit hyperbola.

As we stated on the hyperbolic rotations page, the argument, u, to the cosh() and sinh() functions is twice the light-brown area shown in figure 1 (which we have copied from our hyperbolic rotations page).  We'll prove that here; the proof consists of a simple integration.


Figure 1 -- The unit hyperbola, showing u:



In figure 2 we show the area we need to find, broken down.

Figure 2 -- Area shown in figure 1 is the area on the left, minus the area on the right:

Minus

The area shown in figure 2 is:

(4)    

We substitute:

(5)    

which, applying equation (3), produces:

(6)    

After multiplying out the first term and collecting the pieces, and making a quick trip to the handy-dandy CRC Math Tables for the integral, we obtain:

(7)    

which, after collecting terms, is just:

(8)    

which was to be shown.  

Page created on 11/15/06