Tangent Tangent THC Category="Differential geometry"Category="Differential topology"In Mathematics, the word "tangent" has two distinct, but etymologically related meanings: one in Geometry, and one in Trigonometry.

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.

[image]

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 (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). 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.

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) = 12x − 24; or: y = 12x − 16.

Trigonometry

In Trigonometry, the tangent is a function (see Trigonometric function) defined as:
tan x =  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:
d ( tan x)
––––––––––––––
d x		  =  1
––––––––
cos 2 x		  = sec 2 x.

See also

External links