# Definition:Tangent Function/Definition from Circle/Fourth Quadrant

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## Definition

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.

## Linguistic Note

The name of the **tangent function** derives from its definition as the length of the tangent to a circle.

## Sources

- 1953: L. Harwood Clarke:
*A Note Book in Pure Mathematics*... (previous) ... (next): $\text V$. Trigonometry: Angles larger than $90 \degrees$