Definition:Tangent Function/Definition from Circle/First Quadrant

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


 * TangentFirstQuadrant.png

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

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

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.