tangent - Geometry, Trigonometry, Derivative, Power series

A line (usually a straight line) which touches a curve at a point P with the same gradient as the curve at P. It is sometimes convenient to think of a tangent meeting a curve at two (or more) coincident points. The tangent to a circle at a point P is perpendicular to the radius of the circle through P.

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; The curve, at point P, has the same slope as a tangent passing through P. 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.

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. The curve has a non-vertical tangent at the point (x0, y0) if and only if the function is differentiable at x0. 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. 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.

Given a function and the slope of one of its tangents, we can determine an equation of the tangent line.

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 of a certain segment of a tangent (in the geometric sense) to the unit circle. 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.

The trigonometric tangent function arises as a generating function in combinatorics;

Derivative

The derivative of the tangent is sec²x (using the quotient rule):

Power series

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