# Definition:Tangent Function/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 $A = \tuple {1, 0}$.

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

Then the **tangent** of $\theta$ is defined as the length of $AB$.

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

### Second 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 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 $A = \tuple {1, 0}$.

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

Then the **tangent** of $\theta$ is defined as the length of $AB$.

Hence in the second quadrant, the **tangent** is negative.

### Third 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 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 $A = \tuple {1, 0}$.

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

Then the **tangent** of $\theta$ is defined as the length of $AB$.

Hence in the second quadrant, the **tangent** is negative.

### Fourth 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 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 $A = \tuple {1, 0}$.

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

Then the **tangent** of $\theta$ is defined as the length of $AB$.

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