# Cosecant

A term that represents a ratio of lengths of hypotenuse to opposite side at an angle of a right triangle is called the cosecant.

## Introduction

Cosecant is a name and it is introduced in trigonometry mathematics to denote the ratio of lengths of hypotenuse to opposite side at a particular angle in a right triangle. It is generally written in ratio form and also written as cosecant with angle in alternate way.

#### 1. Ratio

The value of cosecant at an angle is calculated by the ratio of lengths of hypotenuse to opposite side.

$\dfrac{Length \, of \, Hypotenuse}{Length \, of \, Opposite \, side}$

So, cosecant is often called as a trigonometric ratio in general.

#### 2. Function

The value of cosecant at an angle is written in alternative mathematical form by writing cosecant in its short form $\csc$ or $\operatorname{cosec}$ and then respective angle of the right triangle.

For example, if angle of a right angled triangle is represented by $x$, then cosecant of angle $x$ is written as $\csc{x}$ or $\operatorname{cosec}{x}$ in trigonometry. $\csc{x}$ or $\operatorname{cosec}{x}$ is a function form. Hence, it is generally called as cosecant function in mathematics.

Thus, cosecant functions like $\csc{A}$, $\csc{\alpha}$, $\csc{\beta}$, and etc. are defined in trigonometric mathematics.

### Mathematical form

$\Delta CAB$ is a right triangle and its angle is denoted by theta ($\theta$).

cosecant of angle is written as $\csc{\theta}$ or $\operatorname{cosec}{\theta}$ in this case.

$\csc{\theta}$ (or) $\operatorname{cosec}{\theta}$ $\,=\, \dfrac{Length \, of \, Hypotenuse}{Length \, of \, Opposite \, side}$

It is often used as a formula to calculate the value of cosecant at any angle of the right triangle.

$AC$ is length of hypotenuse and $BC$ is length of opposite side (perpendicular) in this example.

$\,\,\, \therefore \,\,\,\,\,\,$ $\csc{\theta}$ (or) $\operatorname{cosec}{\theta}$ $\,=\, \dfrac{AC}{BC}$

#### Values

The list of exact values of cosecant functions in fraction and decimal forms in a table with proofs.

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