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