Definition:Secant Function
This page is about Secant Function in the context of Trigonometry. For other uses, see Secant.
Definition
Definition from Triangle
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.
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 $OB$.
Hence in the first quadrant, the secant is positive.
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
- Definition:Sine
- Definition:Cosine
- Definition:Tangent Function
- Definition:Cotangent
- Definition:Cosecant
- 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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): secant