Well, the Hyperbolic Cosine function, known as the coshine function (or cosh function) IS exactly what you are told it to be (though you have a typo)...

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What you have given is actually the Hyperbolic Sine, known as the shine function (or sinh function) is

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These fall under the family of functions known as the Hyperbolic functions (though many people call them the Alcoholic functions).

The reason they have similar names to the trigonometric functions is because the hyperbolic functions have similar results to those of the trigonometric functions.

Hyperbolic function - Wikipedia, the free encyclopedia