# Equivalence of Definitions of Cosecant of Angle

## Contents

## Theorem

Let $\theta$ be an angle.

The following definitions of the concept of **cosecant** are equivalent:

### Definition from Triangle

In the above right triangle, we are concerned about the angle $\theta$.

The **cosecant** of $\angle \theta$ is defined as being $\dfrac{\text{Hypotenuse}} {\text{Opposite}}$.

### Definition from Circle

Consider a unit circle $C$ whose center is at the origin of a cartesian coordinate plane.

Let $P$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis.

Let a tangent line be drawn to touch $C$ at $A = \left({0, 1}\right)$.

Let $OP$ be produced to meet this tangent line at $B$.

Then the **cosecant** of $\theta$ is defined as the length of $OB$.

## Proof

### Definition from Triangle implies Definition from Circle

Let $\csc \theta$ be defined as $\dfrac {\text{Hypotenuse}} {\text{Opposite}}$ in a right triangle.

Consider the triangle $\triangle OAB$.

By construction, $\angle OAB$ is a right angle.

From Parallelism implies Equal Alternate Interior Angles, $\angle OBA = \theta$.

Thus:

\(\displaystyle \csc \theta\) | \(=\) | \(\displaystyle \frac {OB} {OA}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {OB} 1\) | as $OA$ is the radius of the unit circle | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle OB\) |

That is:

- $\csc \theta = OB$

$\Box$

### Definition from Circle implies Definition from Triangle

Let $\csc \theta$ be defined as the length of $OB$ in the triangle $\triangle OAB$.

Compare $\triangle OAB$ with $\triangle ABC$ in the diagram above.

From Parallelism implies Equal Alternate Interior Angles, $\angle OBA = \theta$.

We have that:

- $\angle CAB = \angle OBA = \theta$
- $\angle ABC = \angle OAB$ which is a right angle

Therefore by Triangles with Two Equal Angles are Similar it follows that $\triangle OAB$ and $\triangle ABC$ are similar.

By definition of similarity:

\(\displaystyle \frac {\text{Hypotenuse} } {\text{Opposite} }\) | \(=\) | \(\displaystyle \frac {AC} {BC}\) | by definition | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {OB} {OA}\) | by definition of similarity | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle OB\) | as $OA$ is the radius of the unit circle | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \csc \theta\) | by definition |

That is:

- $\dfrac {\text{Hypotenuse} } {\text{Opposite} } = \csc \theta$

$\blacksquare$