**Hyperbolic functions** are a special class of transcendental functions, similar to trigonometric functions or the natural exponential function, e^{x}. Although not as common as their trig counterparts, the hyperbolics are useful for some applications, like modeling the shape of a power line hanging between two poles.

Each trigonometric function has a corresponding hyperbolic function, with an extra letter “h”. For example, sinh(x), cosh(x) and tanh(x). While the “ordinary” trig functions parameterize (model) a curve, the hyperbolics model a hyperbola—hence the name.

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## Defining Using e^{(x)}

Unlike their trigonometric counterparts, hyperbolic functions are defined in terms of the exponential function e^{x}. For example, f(x) = cosh(x) is defined by:

And sinh(x) is defined as:

All of the remaining hyperbolic functions (see list below) can be defined in terms of these two definitions.

## Properties of Hyperbolic Functions

Hyperbolic functions can be even or odd functions.

**Even functions**(symmetric about the y-axis): cosh(x) and sech(x),**Odd functions**(symmetric about the origin): All other hyperbolic functions are odd.

Some of these functions are defined for all reals: sinh(x), cosh(x), tanh(x) and sech(x). Two others, coth(x) and csch(x) are undefined at x = 0 because of a vertical asymptote at x = 0.

## Derivatives of Hyperbolic Functions

The derivatives of hyperbolic functions are almost identical to their trigonometric counterparts:

- sinh(x)
^{′}= cosh(x) - cosh(x)
^{′}= sinh(x) - tanh(x)
^{′}= sech^{2}(x) - coth(x)
^{′}= csch^{2}(x) - csch(x)
^{′}= -csch(x) coth(x) - sech(x)
^{′}= sech(x) tanh(x)

## Limits

For x→ ∞, the limits of the hyperbolic functions are:

- lim
_{x → ± ∞}sinh(x) = ± ∞ - lim
_{x → ± ∞}cosh(x) = ∞ - lim
_{x → ± ∞}tanh(x) = ± 1 - lim
_{x → ± ∞}coth(x) = 0 - lim
_{x → ± ∞}csch(x) = 0 - lim
_{x → ± ∞}sech(x) = ± 1

## References

Graph of cosh(x): Desmos calculator.

Stewart. Math 133. Hyperbolic Functions. Retrieved November 24, 2019 from: https://users.math.msu.edu/users/magyar/Math133/6.7-Hyperbolic-Fcns.pdf

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