Definition:Tangent Function/Definition from Circle/Third Quadrant

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 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.

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 $90 \degrees$