# Definition:Cosecant/Definition from Circle

## Definition

### First Quadrant

Consider a unit circle $C$ whose center is at the origin of a cartesian 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 $A = \tuple {0, 1}$.

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

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

Hence in the first quadrant, the **cosecant** is positive.

### Second Quadrant

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

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

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

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

Hence in the second quadrant, the **cosecant** is positive.

### Third Quadrant

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

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

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

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

As $OP$ needs to be produced in the opposite direction to $P$, the **cosecant** is therefore a negative function in the third quadrant.

### Fourth Quadrant

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

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

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

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

Hence in the fourth quadrant, the **cotangent** is negative.