Tangent

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This article is about the mathematical concept of "tangent". For other meanings, see tangent (disambiguation).

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

1 Geometry
- 1.1 Quotation
2 Calculus
3 Trigonometry
4 Derivative
5 Power series
6 See also
7 External links

[edit] 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 called the tangent line (or tangent). The tangent 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 line 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 conic section, such as a circle, the tangent line will intersect the curve at only one point.) It is also possible for a line to be a double tangent, when it is tangent to the same curve at two distinct points. Higher numbers of tangent points are possible as well.

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

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-dimensional manifold.

[edit] Quotation

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

[edit] 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 (x₀, y₀) where y₀ = f(x₀). The curve has a non-vertical tangent at the point (x₀, y₀) if and only if the function is differentiable at x₀. In this case, the slope of the tangent is given by f '(x₀), where f '(x₀) is the derivative of f(x). The curve has a vertical tangent at (x₀, y₀) 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.

Given a function and the slope of one of its tangents, we can determine an equation of the tangent line. For example, an understanding of the power rule will help one determine that the slope of x³, at x = 2, is 12. Using the point-slope equation, one can write an equation for this tangent: y − 8 = 12(x − 2) = 12x − 24; or: y = 12x − 16.

[edit] Trigonometry

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

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

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 centered 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.

Tangent was introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583).

The trigonometric tangent function arises as a generating function in combinatorics; see alternating permutation.