Hyperbolic Functions


The hyperbolic functions enjoy properties similar to the trigonometric functions; their definitions, though, are much more straightforward:

displaymath121

displaymath122

Here are their graphs: the tex2html_wrap_inline125 (pronounce: "kosh") is pictured in red, the tex2html_wrap_inline127 function (rhymes with the "Grinch") is depicted in blue.


As their trigonometric counterparts, the tex2html_wrap_inline125 function is even, while the tex2html_wrap_inline127 function is odd.

Their most important property is their version of the Pythagorean Theorem.

The verification is straightforward:

eqnarray18

While tex2html_wrap_inline135 , tex2html_wrap_inline137 , parametrizes the unit circle, the hyperbolic functions tex2html_wrap_inline139 , tex2html_wrap_inline141 , parametrize the standard hyperbola tex2html_wrap_inline143 , x>1.

In the picture below, the standard hyperbola is depicted in red, while the point tex2html_wrap_inline139 for various values of the parameter t is pictured in blue.


The other hyperbolic functions are defined the same way, the rest of the trigonometric functions is defined:

eqnarray36

tanh x
coth x
sech x
csch x


For every formula for the trigonometric functions, there is a similar (not necessary identical) formula for the hyperbolic functions:

Let's consider for example the addition formula for the hyperbolic cosine function:

Start with the right side and multiply out:

eqnarray61


Try it yourself!

Prove the addition formula for the hyperbolic sine function:

Show that tex2html_wrap_inline153 .

Here is the answer.

Click here to go to the inverse hyperbolic functions.


Helmut Knaust
Fri Jul 19 10:46:10 MDT 1996

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