The **cosecant** function is an old **function** in the area of **mathematics** . It was mentioned in the works of GJ von Lauchen Rheticus in 1596 and E. Gunter around 1620. It was widely used by L. Euler in 1748 and T. Olivier, Wait and Jones in 1881. The classical definition of function cosecant for real arguments is: *“the cosecant of an angle in a right angle triangle is the relationship between the length of the hypotenuse and the length of the opposite angle “*. This description is valid for when this triangle is not degenerate. This approximation can be expanded to arbitrary

**real values**if the arbitrary point in the

**Cartesian plane**is considered and defined as the ratio assuming that α is the value of the angle between the positive

**direction**of the

**axis**and the

**direction**from the origin to the point.

## What is the cosecant?

The **inverse ****function** of the sine of an arc or of an **angle** is known as a cosecant . It is a term widely used in the area of mathematics when it is related to the area of **geometry** .

- Characteristics of the cosecant
- What is it for
- How it is calculated
- Derivative of cosecant
- Integral
- Domain and range
- Inverse function

## Characteristics of the cosecant

The main characteristics of the cosecant are the following:

- It is represented by the letters
**csc**or**cosec**. - It is an
**inverse**trigonometric relationship to the**sine**. - It is known as the
**multiplicative inverse**. - Its domain is
**R – {k · π} with k****∈ Z** - The
**path**of the cosecant is R – (- 1, 1). **It does not cut**the X axis, nor does it**cut**the Y axis.- An important characteristic is that it is odd, which means that it is
**symmetric**with respect to the**origin**: cosec (- x) = – cosec (x). - It also has
**infinite**minima that are relative at points of the form (π / 2 + 2 k π, 1) – with k ∈ Z - It is
**periodic**of period 2π cosec (x) = cosec (x + 2π). - It also has
**vertical****asymptotes**at points of the form x = k π with k∈ - It is not
**limited**.

## What is it for

Like the other trigonometric functions, they are useful, for example, in the field of **astronomy** since it is used to measure distances between stars, in **measuring** distances between geographical points, and in satellite **navigation** systems. They are very functional in the case of the **construction** of houses or buildings to know the measures that must be done. It is also useful in **civil engineering,** to calculate distances, angles of inclination or superelevation of a road. In scientific development it is used in the elaboration of **numerical methods** by mathematicians to make **differential equation** or solve an integral that cannot be worked with conventional methods.

## How it is calculated

The formula to find the cosecant is as follows:

**Csc ****α = 1 / sin α ****= c / a**** **

## Derivative of cosecant

The derivative of the cosecant of a function will be the same unless the cosecant of the function times the cotangent of the function and also the derivative of the function. The formula is as follows:

**F (x) = cosec u**

**F ‘(x) = – (u ㆍ cosec u) **** / (sen² u) = – ****u ****ㆍ ****cosec or ****ㆍ ****cotg or**

## Integral

The formula used for the integral of the cosecant is as follows:

**∫ csc u ****ㆍ du = – log (cot u + csc u)**

This cosecant su can find out when derivation formulas are mastered

## Domain and range

It is important to remember that the domain of a function is made up of all the input values that a **trigonometric** function can carry out. The cosecant domains are **restricted** and can only be used for the functions of **measuring** the **angles** with the output numbers. Every time then that the terminal side that has an angle is along the **X-axis** (where y is equal to 0), the **cosecant function** of the angle cannot be executed .

## Inverse function

In the area of trigonometry, the inverse function of the cosecant is known as the **arcoscant** of an angle. This property is symbolized as: **arccosec ****α** or it can also be identified as **arccsc ****α. ** Its meaning in geometry is the angle whose cosecant is alpha.

**Y = arccsc (x)**

**X = csc (y)**

From this definition we can deduce some important expressions, such as:

**Arccosec (-x) = – arccosec x**

**Arccsc (x) = arcsin (1 / x)**