# Definition:Tangent Function

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## 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:

$\sin x$ is the sine of $x$
$\cos x$ 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 $\tan z$ is defined as:

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

where:

$\sin z$ is the sine of $z$
$\cos z$ 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 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.