# Cotangent

The application of **trigonometric functions** in the area of mathematics and in other areas is widely used in our world. These functions are one of the basic mathematical functions in areas such as **triangulation** , which is used in criminal **investigations** and **cell service** . They are also used in navigation, **surveying** , computer **graphics** , and **music theory** . And one of the most important areas is the **cotangent** trigonometric function .

## What is the cotangent?

The cotangent is equivalent to the result that is equal to the **quotient** between the **adjacent** leg and the **opposite** leg of an **arc** or an **angle** . It is the inverse function of the tangent.

- Characteristics of the cotangent
- What is it for
- How it is calculated
- Formula
- Derivative of the cotangent
- Integral
- Domain and range

## Characteristics of the cotangent

The main characteristics of the cotangent are the following:

- It is the inverse
**trigonometric**ratio of the tangent. - It is the
**reciprocal**or**multiplicative inverse**of the tangent, that is, so α · cot α = 1. - Its abbreviation is
**cot, cotg or cotan**. - Its
**domain**is equal to**R except a · π**, where a is an integer. - Has a
**tour**function is the co-domain of the cotangent, this is equal to**R**. - Its
**graph**is a cotangentoid wave.

## What is it for

It is used in the same way that the sine, cosine and tangent functions are used . It can be used in function of a **right triangle** , using the opposite and adjacent sides of the triangle, or it can be used in function of **the unit circle** , which shows the angles in radians.

If you want to find the **height of** a particular tree based on the shadow it casts when the sun is at an angle of 30 degrees, we can find this information using cotangent using a 30-60-90 triangle.

It also serves to **model** real situations for physical phenomena that must describe waves such as **sound** , simple **harmonic movement** and others.

## How it is calculated

It can be calculated as follows:

**Cotangent = Adjacent leg / Opposite leg**

Since it is the **inverse** function of the tangent, it can also be calculated by **dividing** 1 by the tangent.

## Formula

The formula to find the cotangent is:

**Cot α = 1 / tan α = contiguous leg / opposite leg = b / a**

## Derivative of the cotangent

The derivative of the cotangent of a given function is equal to the derivative of the function with negative sign divided by the **sinus** to the **square** of this function:

**F (x) = cotg w**

**F´ (x) = (- w´) / (sin² w)**

Applying the trigonometric rules, the derivative of the cotangent can also be defined as follows:

**F´ (x) = – w´ · cosec² w**

**F´ (x) = – w´ · (1 + cotg² w)**

## Integral

The formula used to find the cotangent integral is as follows:

**∫ ****csc x sec x dx**

## Domain and range

The tangent and the cotangent are related not only by the fact that they are reciprocal, but also by the behavior of their **ranges** . In reference to the coordinate plane, tangent is y / X, and cotangent is X / y.

The **domains** of these two functions are **restricted** , because on some occasions their relationships could have zeros in the denominator , but their ranges are infinite.

To find out the domain of the cotangent we must know its value and then proceed to divide:

**cos (x) / sin (x)**

It is important to mention that sin (x) should never take the value of the number 0.

The cotangent domain comprises the group of real numbers minus the multiples that π has so that K is a then an integer.