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.