What is the Relation between tan and tanh?

Date: 08/25/2001 at 03:41:28
From: Barot Ketan
Subject: What is the relation between tan and tanh?

Hello sir,

I want to know the equation type relation between tan and tanh - I 
mean the relation between tangent and hyperbolic tangent.

Date: 08/25/2001 at 05:59:59
From: Doctor Floor
Subject: Re: What is the relation between tan and tanh?

Hello and thanks for writing,

By definition we have:

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

See, from the Dr. Math archives: 

   Hyperbolic Functions
   http://mathforum.org/dr.math/problems/weber10.14.98.html   

Also we have Euler's Equation - see from the Dr. Math FAQ:

   http://mathforum.org/dr.math/faq/faq.euler.equation.html   

    ix
   e   = cos x + i*sin x

(where i is the imaginary unit "sqrt(-1)").

This gives

               ix     -ix                 ix     -ix
              e   -  e                   e   +  e
    sin(x) =  -----------  and cos(x) =  -----------
                  2i                         2

Combination gives
                   ix     -ix
              1    e   -  e       1 sinh(ix)   tanh(ix)
    tan(x) =  - *  ----------- =  - -------- = --------
              i      ix    -ix    i cosh(ix)      i
                   e  +  e

From which we conclude the identity

   i*tan(x) = tanh(ix)

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/