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 $$\frac{\text{Opposite}} {\text{Adjacent}} $$.

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:


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

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

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

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

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


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

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

The definition is valid for all $$z \in \C$$ 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.