# Definition:Tangent Function

## Definition

### Definition from Triangle

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

The **tangent** of $\angle \theta$ is defined as being $\dfrac{\text{Opposite}} {\text{Adjacent}}$.

### 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 **tangent** of $\theta$ is defined as the length of $AB$.

Hence in the first quadrant, the **tangent** is positive.

### Real Function

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

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

- $\tan x = \dfrac {\sin x} {\cos x}$

where:

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 $\tan z$ is defined as:

- $\tan z = \dfrac {\sin z} {\cos z}$

where:

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

## Also see

- Definition:Sine
- Definition:Cosine
- Definition:Cotangent
- Definition:Secant Function
- Definition:Cosecant

- Results about
**the tangent function**can be found**here**.

## Linguistic Note

The word **tangent** comes from the Latin **tangentus** *that which is touching*, the present participle of **tangere** *to touch*.

It is pronounced with a soft g: ** tan-jent**.

## Sources

- 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles: Trigonometry - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**tangent**