Definition:Secant Function

From ProofWiki
Jump to navigation Jump to search

This page is about Secant Function in the context of Trigonometry. For other uses, see Secant.

Definition

Definition from Triangle

SineCosine.png

In the above right triangle, we are concerned about the angle $\theta$.

The secant of $\angle \theta$ is defined as being $\dfrac{\text{Hypotenuse}} {\text{Adjacent}}$.


Definition from Circle

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


SecantFirstQuadrant.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 $A = \tuple {1, 0}$.

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


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


Real Function

Let $x \in \R$ be a real number.

The real function $\sec x$ is defined as:

$\sec x = \dfrac 1 {\cos x}$

where $\cos x$ is the cosine of $x$.


The definition is valid for all $x \in \R$ such that $\cos x \ne 0$.


Complex Function

Let $z \in \C$ be a complex number.

The complex function $\sec z$ is defined as:

$\sec z = \dfrac 1 {\cos z}$

where $\cos z$ is the cosine of $z$.


The definition is valid for all $z \in \C$ such that $\cos z \ne 0$.


Also see

  • Results about the secant function can be found here.


Linguistic Note

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.


Sources