Definition:Tangent

Tangent Line
From Euclid's, Book III Definitions: 2:


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


 * TangentCircle.png

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

Tangent Circles
From Euclid's, Book III Definitions: 3:


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


 * TangentCircles.png

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

Trigonometry

 * SineCosine.png

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.

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
 * Nature of Tangent Function