# Hyperbolic Functions

The two basic hyperbolic functions are:

sinh and cosh

*(pronounced "shine" and "cosh")*

sinh x = \frac{e^{x} − e^{−x}}{2}

cosh x = \frac{e^{x} + e^{−x}}{2}

They are not the same as sin(x) and cos(x), but are a little bit similar:

*sinh vs sin*

*cosh vs cos*

## Catenary

One of the interesting uses of Hyperbolic Functions is the curve made by suspended cables or chains.

A hanging cable forms a curve called a **catenary** defined using the **cosh** function:

f(x) = a cosh(x/a)

Like in this example from the page arc length :

## Other Hyperbolic Functions

From **sinh** and **cosh** we can create:

### Hyperbolic tangent "tanh" *(pronounced "than")*:

tanh x = \frac{sinh x}{cosh x} = \frac{e^{x} − e^{−x}}{e^{x} + e^{−x}}

*tanh vs tan*

### Hyperbolic cotangent:

coth x = \frac{cosh x}{sinh x} = \frac{e^{x} + e^{−x}}{e^{x} − e^{−x}}

### Hyperbolic secant:

sech x = \frac{1}{cosh x} = \frac{2}{e^{x} + e^{−x}}

### Hyperbolic cosecant "csch" or "cosech":

csch x = \frac{1}{sinh x} = \frac{2}{e^{x} − e^{−x}}

## Why the Word "Hyperbolic" ?

Because it comes from measurements made on a Hyperbola:

So, just like the trigonometric functions relate to a circle, the hyperbolic functions relate to a hyperbola.## Identities

- sinh(−x) = −sinh(x)
- cosh(−x) = cosh(x)

And

- tanh(−x) = −tanh(x)
- coth(−x) = −coth(x)
- sech(−x) = sech(x)
- csch(−x) = −csch(x)

## Odd and Even

Both **cosh** and **sech** are Even Functions, the rest are Odd Functions.

## Derivatives

Derivatives are:

\frac{d}{dx} sinh x = cosh x

\frac{d}{dx} cosh x = sinh x

\frac{d}{dx} tanh x = 1 − tanh^{2} x