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# Hyperbolic Functions

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

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

As their trigonometric counterparts, the  function is even, while the  function is odd.
Their most important property is their version of the Pythagorean Theorem.

The verification is straightforward:
While  ,  , parametrizes the unit circle, the hyperbolic functions  ,  , parametrize the standard hyperbola  , x>1.
In the picture below, the standard hyperbola is depicted in red, while the point  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:

 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: