TANGENT

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

Contents

Geometry

Quotation

Calculus

Trigonometry

Derivative

Integral

Power series of the tangent function

See also

External links

Geometry

In plane geometry, a 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. 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.

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)

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), where ''f'' '(''x'') is the derivative of f(x). The curve has a vertical tangent at (''x''₀, ''y''₀) if and only if the slope of the secant lines approaches plus or minus infinity as one approaches the point from either side.
The secant lines can be used to approximate the tangent; informally, the slope of a secant "approaches" the slope (or direction) of the tangent, as the secants' "other" point approaches the first one. The problem of finding the tangent line to a graph or the 'tangent line problem' was one of the main problems that originated calculus, in calculus this problem is solved using Newton's difference quotient. Newton's and Leibniz' original definitions were criticized for not being precise. Today, concepts like "approaches" are usually made rigorous via the definition of limit.
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 ''x''³ (as the derivative of x³ would be 3''x''²), at ''x'' = 2, is 12. Using the point-slope equation, one can write an equation for this tangent: ''y'' − 8 = 12(''x'' − 2); ''y'' − 8 = 12''x'' − 24; or y = 12''x'' − 16.

Trigonometry

In trigonometry, the 'tangent' is a function (see trigonometric function) defined as
:
The function is so-named because it can be defined as the length 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 either intersect the tangent line at a single point Q, or not at all. 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.

Derivative

The derivative of the tangent is found by using the quotient rule:
:

Integral

The antiderivative of the tangent function is given by:
:$int\; an\; x,dx\; =\; -ln(|cos\; x|)\; +\; C,.$
This can be shown by taking the derivative of the right-hand side, using the chain rule:
:

Power series of the tangent function

:
See also the list of Taylor series of some common functions.

See also

★ Secant method

★ Subtangent

★ Tangential component

★ Law of tangents

★ List of trigonometric identities

External links

★ MathWorld: Tangent

★ MathWorld: Tangent Line

★ Tangent to a circle With interactive animation

★ Tangent and first derivative - An interactive simulation

★ The Tangent Parabola by John H. Mathews