Definition:Tangent Function

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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:

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


$\sin x$ 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:

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


$\sin z$ 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.