# 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**