# Hyperbolic Functions

The two basic hyperbolic functions are "sinh" and "cosh":

### Hyperbolic Sine:

sinh(x) = \frac{e^{x} − e^{-x}}{2}

*(pronounced "shine")*

### Hyperbolic Cosine:

cosh(x) = \frac{e^{x} + e^{-x}}{2}

*(pronounced "cosh")*

They use the natural exponential function **e ^{x}**

And are not the same as sin(x) and cos(x), but 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

The 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)