Definition:Tangent

Definition

Geometry

Tangent Line

As Euclid defined it:

A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.

(The Elements: Book III: Definition $2$)

In the above diagram, the line is tangent to the circle at the point $C$.

Tangent Circles

As Euclid defined it:

Circles are said to touch one another which, meeting one another, do not cut one another.

(The Elements: Book III: Definition $3$)

In the above diagram, the two circles are tangent to each other at the point $C$.

Trigonometry

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

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

Thus it is seen that the tangent is the sine over the cosine.

Analysis

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 $\cos z \ne 0$.

Linguistic Note

The word tangent comes from the Latin tango, tangere (I touch, to touch).

Also see

• Sine, cosine, cotangent, secant and cosecant
• Shape of Tangent Function