Tangent
From Academic Kids

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 straightline 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 nontangential 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 higherdimensional 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 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 nonvertical 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 pointslope 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:
 <math>\,\tan x = \frac{\,\sin x}{\,\cos x}<math>
The function is sonamed 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 twodimensional 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 xaxis, 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:
 <math>\,\frac{\mathrm{d} (\tan x)}{\mathrm{d} x} = \frac{1}{\cos^2 x}=\sec^2 x.<math>
See also
External link
 MathWorld: Tangent (http://mathworld.wolfram.com/Tangent.html)da:Tangent
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