# Tangent

In mathematics, the word "**tangent**" has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry.

## Contents

## Geometry

In plane geometry, a straight line is **tangent** to a curve, at some point, if both line and curve pass through the point with the same direction; such a line is the best straight-line approximation to the curve at that point. The curve, at point *P*, has the same slope as a tangent passing through *P*. The slope of a **tangent line** can be approximated by a secant line. It is a mistake to think of tangents as lines which intersect a curve at only one single point. There are tangents which intersect curves at several points (as in the following example), and there are non-tangential lines which intersect curves at only one single point. (Note that in the important case of a circle, however, the tangent line will intersect the curve at only one point.)

In the following diagram, a red line intersects the black curve at two points. It is tangent to the curve at one point.

In higher-dimensional geometry, one can define the **tangent plane** for a surface in an analogous way to the tangent line for a curve. In general, one can have an (*n* − 1)-dimensional **tangent hyperplane** to an (*n* − 1)-dimensional manifold.

### Quote

"And I dare say that this is not only the most useful and general [concept] in geometry, that I disproved, but even that I ever desired to disprove." Descartes (1637)

### Related Meaning

To "go off on a tangent" means to go off topic, likely derived from the geometric meaning of a tangent.

## Calculus

A "formal" definition of the tangent requires calculus. Specifically, suppose a curve is the graph of some function, *y* = *f*(*x*), and we are interested in the point (*x*_{0}, *y*_{0}) where *y*_{0} = *f*(*x*_{0}). The curve has a non-vertical tangent at the point (*x*_{0}, *y*_{0}) if and only if the function is differentiable at *x*_{0}. In this case, the slope of the tangent is given by *f* '(*x*_{0}). The curve has a vertical tangent at (*x*_{0}, *y*_{0}) if and only if the slope approaches plus or minus infinity as one approaches the point from either side.

Above, it was noted that a secant can be used to approximate a tangent; it could be said that the slope of a secant approaches the slope (or direction) of the tangent, as the secants' points of intersection approach each other. Should one also understand the notion of a limit; one might understand how that concept is applicable to those discussed here, via calculus. In essence, calculus was developed (in part) as a means to find the slopes of tangents; this challenge, being known as the **tangent line problem**, is solvable via Newton's difference quotient.

Should one know the slope of a tangent, to some function; then, one can determine an equation for the tangent. For example, an understanding of the power rule will help one determine that the slope of *x*^{3}, at *x* = 2, is 12. Using the point-slope equation, one can write an equation for this tangent: *y* − 8 = 12(*x* − 2) = 12*x* − 24; or: y = 12*x* − 16.

## Trigonometry

In trigonometry, the **tangent** is a function (see trigonometric function) defined as:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \,\tan x = \frac{\,\sin x}{\,\cos x}}**

The function is so-named because it can be defined as the length of a certain segment of a tangent (in the geometric sense) to the unit circle. It is easiest to define it in the context of a two-dimensional Cartesian coordinate system. If one constructs the unit circle centred at the origin, the tangent line to the unit circle at the point P = (1, 0), and the ray emanating from the origin at an angle θ to the *x*-axis, then the ray will intersect the tangent line at *at most* a single point Q. The tangent (in the trigonometric sense) of θ is the length of the portion of the tangent line between P and Q. If the ray does not intersect the tangent line, then the tangent (function) of θ is undefined.

### Derivative

The derivative of the tangent is:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \,\frac{\mathrm{d} (\tan x)}{\mathrm{d} x} = \frac{1}{\cos^2 x}=\sec^2 x.}**

## See also

## External links

da:Tangent de:Tangente de:Tangens es:Tangente fr:Tangente he:משיק pl:Styczna sv:Tangent zh:切线