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.

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

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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.

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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

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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:

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Start with the right side and multiply out:

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Try it yourself!

Prove the addition formula for the hyperbolic sine function:Show that  .