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In mathematics, the word "tangent" has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ...
In historical linguistics, etymology is the study of the origins of words. ...
Geometry (from the Greek words Geo = earth and metro = measure) is the branch of mathematics first popularized in ancient Greek culture by Thales (circa 624-547 BC) dealing with spatial relationships. ...
Trigonometry (from the Greek trigonon = three angles and metro = measure) is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine, cosine and tangent. ...
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 mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...
A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ...
In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...
The word point can refer to: a location in physical space a unit of angular measurement; see navigation point is a typographic unit of measure in typography equal inch or sometimes approximated as inch; on computer displays it should be equal to point in typography if the correct display resolution...
A secant line of a curve is a line that intersects two (or more) points on the curve. ...
In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre. ...
In the following diagram, a red line intersects the black curve at two points. It is tangent to the curve at one point.
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 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. A tangent to a curve, created with GIMP File links The following pages link to this file: Tangent Categories: GFDL images ...
In mathematics, a surface is a two-dimensional manifold. ...
This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ...
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) René Descartes René Descartes (IPA: , March 31, 1596–February 11, 1650), also known as Cartesius, was a French philosopher, mathematician and part-time mercenary. ...
Events February 3 - Tulipmania collapses in Netherlands by government order February 15 - Ferdinand III becomes Holy Roman Emperor December 17 - Shimabara Rebellion erupts in Japan Pierre de Fermat makes a marginal claim to have proof of what would become known as Fermats last theorem. ...
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. For other uses of the term calculus see calculus (disambiguation) Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ...
In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...
↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include Q is necessary and sufficient for P and P...
In mathematics, the derivative of a function is one of the two central concepts of calculus. ...
In mathematics, the slope (or gradient, especially where three or more dimensions are discussed) of a straight line (within a Cartesian coordinate system) is a measure for the steepness of said line. ...
Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. ...
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. Secant can refer to: a secant line secant, a trigonometric function, equivalent to sec(x) = 1/cos(x) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ...
For other uses of the term calculus see calculus (disambiguation) Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ...
Sir Isaac Newton in Godfrey Knellers 1689 portrait Sir Isaac Newton (25 December 1642 – 20 March 1727 by the Julian calendar in use in England at the time; or 4 January 1643 – 31 March 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemist who...
The derivative in mathematics (specifically, differential calculus) is a quantity that measures, on continuous functions, the limit of a rate of change, , as approaches 0. ...
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 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 In mathematics, polynomials are perhaps the simplest functions with which to do calculus. ...
In mathematics, the slope (or gradient, especially where three or more dimensions are discussed) of a straight line (within a Cartesian coordinate system) is a measure for the steepness of said line. ...
In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ...
Trigonometry In trigonometry, the tangent is a function (see trigonometric function) defined as: Trigonometry (from the Greek trigonon = three angles and metro = measure) is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine, cosine and tangent. ...
In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
 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. In general English usage, length (symbols: l, L) is but one particular instance of distance – an objects length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with distance. Height is vertical distance; width (or breadth...
Illustration of a unit circle. ...
Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
Ray has several meanings. ...
Derivative The derivative of the tangent is: 
See also In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. ...
In geometry, the subtangent is the projection of the tangent upon the axis of abscissas (i. ...
For the authentication, authorisation, and accounting protocol, see DIAMETER. In geometry, a diameter (Greek words diairo = divide and metro = measure) of a circle is any straight line segment that passes through the center and whose endpoints are on the circular boundary, or, in more modern usage, the length of such...
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