Definition:Tangent

Geometry
From Euclid's "The Elements":

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



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

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



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

Trigonometry
Let $$\theta$$ be an angle.

The tangent of $$\theta$$, written $$\tan \theta$$ (pronounced "tan theta") is defined as:

$$\tan \theta = \frac {\sin \theta} {\cos \theta}$$

where:
 * $$\sin \theta$$ is the sine of $$\theta$$;
 * $$\cos \theta$$ is the cosine of $$\theta$$.

The definition is valid for all angles where $$\cos \theta \ne 0$$.

Real Function
Let $$x \in \mathbb{R}$$ be a real number.

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

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

where:
 * $$\sin x$$ is the sine of $$x$$;
 * $$\cos x$$ is the cosine of $$x$$.

The definition is valid for all $$x \in \mathbb{R}$$ such that $$\cos x \ne 0$$.

Complex Function
Let $$z \in \mathbb{C}$$ be a complex number.

The complex function $$\tan z$$ is defined as:

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

where:
 * $$\sin z$$ is the sine of $$z$$;
 * $$\cos z$$ is the cosine of $$z$$.

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

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