In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. The word tangent comes from a Latin word that means touching, and has several meanings: A clavichord tangent is a part of the action of the clavichord that both initates and sustains a tone, and helps determine pitch. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Not to be confused with Entomology, the scientific study of insects. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with...

Geometry

Tangent line to a circle (as opposed to a secant).

In plane geometry, a 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 called the tangent line (or tangent). The tangent 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 line 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 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. Higher numbers of tangent points are possible. 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. Image File history File links CIRCLE_LINES.svg Summary A circle showing the chord, secant, and tangent. ... Image File history File links CIRCLE_LINES.svg Summary A circle showing the chord, secant, and tangent. ... In mathematics, plane geometry may mean: geometry of the Euclidean plane; or sometimes geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others; or geometry of the hyperbolic plane or two-dimensional spherical geometry. ... A representation of one line Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... Look up Slope in Wiktionary, the free dictionary. ... A secant line of a curve is a line that intersects two or more points on the curve. ... Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... Circle illustration This article is about the shape and mathematical concept of circle. ...

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-dimensional manifold. An open surface with X-, Y-, and Z-contours shown. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ...

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) René Descartes (French IPA: ) (March 31, 1596 – February 11, 1650), also known as Renatus Cartesius (latinized form), was a highly influential French philosopher, mathematician, scientist, and writer. ... 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

Tangent to a general curve.

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 '(x) is the derivative of f(x). The curve has a vertical tangent at (x₀, y₀) if and only if the slope of the secant lines approaches plus or minus infinity as one approaches the point from either side. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... Look up Slope in Wiktionary, the free dictionary. ... For a non-technical overview of the subject, see Calculus. ... For a non-technical overview of the subject, see Calculus. ... The infinity symbol ∞ in several typefaces. ...

The secant lines can be used to approximate the tangent; informally, the slope of a secant "approaches" the slope (or direction) of the tangent, as the secants' "other" point approaches the first one. The problem of finding the tangent line to a graph or the tangent line problem was one of the main problems that originated calculus, in calculus this problem is solved using Newton's difference quotient. Newton's and Leibniz' original definitions were criticized for not being precise. Today, concepts like "approaches" are usually made rigorous via the definition of limit. A secant line of a curve is a line that intersects two or more points on the curve. ... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... For a non-technical overview of the subject, see Calculus. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as...

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 (as the derivative of x³ would be 3x²), at x = 2, is 12. Using the point-slope equation, one can write an equation for this tangent: y − 8 = 12(x − 2); y − 8 = 12x − 24; or y = 12x − 16. In mathematics, polynomials are perhaps the simplest functions with which to do calculus. ... Look up Slope in Wiktionary, the free dictionary. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ...

Trigonometry

The graph of the tangent function.

In trigonometry, the tangent is a function (see trigonometric function) defined as Image File history File links Tan. ... Image File history File links Tan. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ...

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 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. Look up length, width, breadth in Wiktionary, the free dictionary. ... Illustration of a unit circle. ... Fig. ... A representation of one line Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ...

Tangent was introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583). Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Thomas Fincke (January 6, 1561 - April 24, 1656) was a Danish mathematician and physicist, and a professor at the University of Copenhagen for more than sixty years. ... 1583 was a common year starting on Saturday of the Gregorian calendar or a common year starting on Tuesday of the Julian calendar. ...

The trigonometric tangent function arises as a generating function in combinatorics; see alternating permutation. In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ... In combinatorial mathematics, an alternating permutation of the set {1, 2, 3, ..., n} is an arrangement of those numbers into an order c1, ..., cn such that no element ci is between ci âˆ’ 1 and ci + 1 for any value of i. ...

Derivative

The derivative of the tangent is found by using the quotient rule: In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist. ...

Integral

The antiderivative of the tangent function is given by: In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...

This can be shown by taking the derivative of the right-hand side, using the chain rule: In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

Power series of the tangent function

See also the list of Taylor series of some common functions. As the degree of the Taylor series rises, it approaches the correct function. ...

See also

• Secant method
• Subtangent
• Tangential component
• Law of tangents
• List of trigonometric identities

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. ... Illustration of tangential and normal components of a vector to a surface. ... In trigonometry, the law of tangents is a statement about arbitrary triangles in the plane. ... In mathematics, trigonometric identities are equalities that involve trigonometric functions that are true for all values of the occurring variables. ...

External links

• MathWorld: Tangent
• MathWorld: Tangent Line
• Tangent to a circle With interactive animation
• Tangent and first derivative - An interactive simulation
• The Tangent Parabola by John H. Mathews