Definition:Tangent Function/Definition from Circle/Third Quadrant

Definition
Consider a unit circle $C$ whose center is at the origin of a cartesian coordinate plane.


 * TangentThirdQuadrant.png

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 $A = \left({1, 0}\right)$.

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

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

Linguistic Note
Hence the name of this tangent function: it is the length of the tangent line as described.