# Trigonometry/Cosh, Sinh and Tanh

The functions **cosh x**, **sinh x** and **tanh x** have much the same relationship to the rectangular hyperbola y^{2} = x^{2} - 1 as the circular functions do to the circle y^{2} = 1 - x^{2}. They are therefore sometimes called the **hyperbolic functions** (*h* for *hyperbolic*).

Notation and pronunciation
is an abbreviation for 'cosine hyperbolic', and is an abbreviation for 'sine hyperbolic'. is pronounced is pronounced 'cosh', as you'd expect, and is pronounced |

[Diagram of rectangular hyperbola to illustrate]

## Contents

## Definitions[edit]

They are defined as

Equivalently,

Reciprocal functions may be defined in the obvious way:

1 - tanh^{2}(x) = sech^{2}(x); coth^{2}(x) - 1 = cosech^{2}(x)

It is easily shown that , analogous to the result In consequence, sinh(x) is always less in absolute value than cosh(x).

sinh(-x) = -sinh(x); cosh(-x) = cosh(x); tanh(-x) = tanh(x).

Their ranges of values differ greatly from the corresponding circular functions:

- cosh(x) has its minimum value of 1 for x = 0, and tends to infinity as x tends to plus or minus infinity;
- sinh(x) is zero for x = 0, and tends to infinity as x tends to infinity and to minus infinity as x tends to minus infinity;
- tanh(x) is zero for x = 0, and tends to 1 as x tends to infinity and to -1 as x tends to minus infinity.

[Add graph]

## Addition formulae[edit]

There are results very similar to those for circular functions; they are easily proved directly from the definitions of cosh and sinh:

- sinh(x+y) = sinh(x)cosh(y) + cosh(x)sinh(y)

- cosh(x+y) = cosh(x)cosh(y) + sinh(x)sinh(y)

## Inverse functions[edit]

If y = sinh(x), we can define the inverse function x = sinh^{-1}y, and similarly for cosh and tanh. The inverses of sinh and tanh are uniquely defined for all x. For cosh, the inverse does not exist for values of y less than 1. For y = 1, x = 0. For y > 1, there will be two corresponding values of x, of equal absolute value but opposite sign. Normally, the positive value would be used. From the definitions of the functions,

## Simplifying a cosh(x) + b sinh(x)[edit]

If a > |b| then

- where :

If |a| < b then

- where :

## Relations to complex numbers[edit]

- cos(
*i*x) = cosh(x) - sin(
*i*x) =*i*sinh(x) - tan(
*i*x) =*i*tanh(x)

The addition formulae and other results can be proved from these relationships.

## The gudermannian[edit]

The **gudermannian** (named after Christoph Gudermann, 1798–1852) is defined as gd(x) = tan^{-1}(sinh(x)). We have the following properties:

- gd(0) = 0;
- gd(-x) = -gd(x);
- gd(x) tends to
^{1}⁄_{2}π as x tends to infinity, and -^{1}⁄_{2}π as x tends to minus infinity.

The inverse function gd^{-1}(x) = sinh^{-1}(tan(x)) = ln(sec(x)+tan(x)).

## Differentiation[edit]

As can be proved from the definitions above,

We also have

.