# How to use the ACOTH function

**What is the ACOTH function?**

The ACOTH function calculates the inverse hyperbolic cotangent of a number.

**What is the trigonometric tangent?**

The trigonometric tangent is one of the basic trigonometric functions. It is defined as the ratio between the opposite side and adjacent side lengths in a right triangle.

tan Î¸ = opposite / adjacent

The tangent relates an angle to the side lengths in a right triangle.

TAN function | Tangent Function - CueMath

**What is the cotangent?**

The cotangent is a trigonometric function that is closely related to the tangent, abbreviated cot.

It is defined as the reciprocal of the tangent:

cot Î¸ = 1 / tan Î¸

or

cot Î¸ = adjacent / opposite

COT function | Cotangent math formula

**What is the hyperbolic cotangent?**

The hyperbolic cotangent, abbreviated coth, is one of the hyperbolic trigonometric functions.

Defined as the reciprocal of the hyperbolic tangent:

coth(x) = 1 / tanh(x)

or using e which is the base of the natural logrithm:

coth(x) = (e^x + e^-x) / (e^x - e^-x)

**What is hyperbolic?**

The term hyperbolic refers to things related to hyperbolas.

**What is hyperbolas?**

A hyperbola is a type of conic section curve.

The hyperbolas can be defined by the equations:

x^{2} - y^{2} = 1 returns a hyperbola centered at the origin that opens left and right.

and

x^{2} - y^{2} = -1 returns a hyperbola at the origin, but opening up and down.

**What is the difference between the hyperbolic functions and the trigonometric functions?**

Hyperbolic functions:

- take a real number input
- defined in terms of the exponential function e^x
- defined for all real numbers
- not periodic
- graphs have two asymptotes

Trigonometric functions:

- take an angle as input
- defined based on angles and the circle unit
- have a periodic domain
- do not have asymptotes
- repeat every 2Ï€ radians
- do not have asymptotes

**What is the inverse hyperbolic cotangent?**

The inverse hyperbolic cotangent finds the hyperbolic angle whose cotangent is a given value x, abbreviated to coth^{-1}

**What is the angle Î¸?**

The Greek letter theta (Î¸) is commonly used to represent an unknown angle.

Hyperbolic functions - Wikipedia

**What is reciprocal of a number?**

The reciprocal of a number is 1 divided by that number. For example, if x isÂ a number then 1/x is the reciprocal.

**What are asymptotes in terms of hyperbolic function?**

Asymptotes refer to straight lines that the function curves towards but never intersects.

#### Table of Contents

## 1. ACOTH function syntax

ACOTH(*number*)

## 2. ACOTH function arguments

number |
Required. The absolute value of Number must be greater than 1. |

### Comments

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## 3. ACOTH function example

Formula in cell C3:

## 4. ACOTH function not working

ACOTH returns a #NUM! error ff the specified number is less than 1.

## 5. How is the ACOTH function calculated?

ACOTH(x) = 0.5*LN((x+1)/(x-1))

### Useful links

### Functions in 'Math and trigonometry' category

The ACOTH function function is one of 73 functions in the 'Math and trigonometry' category.

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