Hyperbolic Functions

Date: 10/14/98 at 00:04:55
From: Anne Weber
Subject: Trig Functions using exponentials

A. What do cosh(x) and sinh(x) stand for? Why are they called this?  
   These functions occur naturally in the physical sciences. Give one 
   example of where they might occur.

B. What is the value of sinh(2)?

C. What's the definition of cosh(x) and sinh(x) in terms of 
   exponential functions?

D. Use you answer in part C to prove that tanh^2(x) + sech^2(x) = 1 if:

             sinh(x)                   1
   tanh(x) = -------  and sech(x) = -------
             cosh(x)                cosh(x)

Thanks in advance,
Anne Weber

Date: 10/14/98 at 04:37:02
From: Doctor Pat
Subject: Re: Trig Functions using exponentials

Anne,

Cosh(x) (pronounced so it rhymes with "gosh") and sinh(x) (sounds like 
cinch, or some say sink) are called hyperbolic functions because they 
can be defined on a unit hyperbola, just as the others are called 
circular functions, because they are defined on a unit circle. An 
example is a hanging cable, the catenary arch.

Sinh(2) is easily found from the definition below, so I will use it 
there. The definitions are:

                x       -x                  x      -x
               e    -  e                   e   +  e
   sinh(x) =  ------------  and cosh(x) = -----------
                   2                           2

Then when x = 2, we get sinh(2) = (e^2 - e^(-2))/2 which is 
approximately 3.62.

I'm going to leave the last part for you to do. You should be able to 
use the definitions to get this last piece. Just use the definitions 
and substitute in the exponentials and then simplify. 

Good luck,

- Doctor Pat, The Math Forum
  http://mathforum.org/dr.math/