On-line Math 21

3.3  Hyperbolic functions

3.3.1  Basic definitions, hyperbolic trigonometry

Hyperbolic functions sound like they ought to be some great new idea, but they aren't. There are certain rules that they satisfy similarly to rules for trig functions (with a twist), but it's not really a big deal. So they satisfy identities similar to trig functions? That doesn't make them as important as trig functions.

However, you will see these things mentioned from time to time, so we so have to get the definitions straight.

Definition 1 The hyperbolic sine function is the function sinh(x) , which is expressed as:

sinh(x) : = 1 2 (ex-e-x).

That's it. The hyperbolic cosine function is the function coshx which is expressed as:

cosh(x) : = 1 2 (ex+e-x).

The domain of either function is the whole real line, but the range is a bit more subtle to see. For x large, sinh(x) will also be large, and for x ``large and negative'' ( -x is then ``large''), sinh(x) will also be large in absolute value, but negative. So, since sinh(x) is continuous, the range of sinh(x) is the whole real line. On the other hand, since cosh(x) = (e^x+e^-x)/2, it is never negative, and in fact is smallest when x = 0, cosh(x) ³ 1 , and in fact the range is { y| y ³ 1 } .

Now, the rest of the hyperbolic functions are defined as in real trigonometry: tanh(x): = sinh(x)/cosh(x) , sech(x): = 1/cosh(x) , etc.

There are formulas similar to trig functions that do hold:

cosh2(x)-sinh2(x) = 1, cosh(x+y) = cosh(x) cosh(y)+sinh(x) sinh(y), sinh(x+y) = sinh(x) cosh(y)+cosh(x) sinh(y).

There are, as with standard trigonometry, a large number of identities. Most of them are remarkably similar to trigonometric identities, except, as with these, for some changes in sign.

If this isn't enough, there are inverses, too. However, from the point of view of calculus, even the trigonometric inverses weren't all that interesting, except that the derivatives were more basic functions. So, these shouldn't be interesting (in calculus) either, unless they provide new formulas. But they don't (at least, none that can't be easily found other ways). However, there is an odd collection of results: these inverses can be written easily in terms of functions we already know, such as:

sinh-1(x)  = ln(x+ Ö   x2+1   ),

which follows from the definitions.

Explanation

Copyright (c) 2000 by David L. Johnson.