Definition:Secant Function/Definition from Circle

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 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 secant of $\theta$ is defined as the length of $OB$.

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 secant of $\theta$ is defined as the length of $OB$.

As $OP$ needs to be produced in the opposite direction to $P$, the secant is therefore a negative function in the second quadrant.

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 secant of $\theta$ is defined as the length of $OB$.

As $OP$ needs to be produced in the opposite direction to $P$, the secant is therefore a negative function in the third quadrant.

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 secant of $\theta$ is defined as the length of $OB$.

The word secant comes from the Latin secantus (that which is cutting), the present participle of secare to cut.

This arises from the fact that it is the length of the line from the origin which cuts the tangent line.

It is pronounced with the emphasis on the first syllable and a long e: see-kant.