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Secant line
 
 
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A secant line of a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 is a line that (locally) intersects two point
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
s on the curve. The word secant comes from the Latin
Latin

Latin is an Italic language, historically spoken in Latium and Ancient Rome. Through the Military history of the Roman Empire, Latin spread throughout the Mediterranean and a large part of Europe....
 secare, for to cut.

It can be used to approximate the tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
 to a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
, at some point P. If the secant to a curve is defined by two point
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
s, P and Q, with P fixed and Q variable, as Q approaches P along the curve, the direction of the secant approaches that of the tangent at P, assuming there is just one.






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Circle Lines
A secant line of a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 is a line that (locally) intersects two point
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
s on the curve. The word secant comes from the Latin
Latin

Latin is an Italic language, historically spoken in Latium and Ancient Rome. Through the Military history of the Roman Empire, Latin spread throughout the Mediterranean and a large part of Europe....
 secare, for to cut.

It can be used to approximate the tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
 to a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
, at some point P. If the secant to a curve is defined by two point
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
s, P and Q, with P fixed and Q variable, as Q approaches P along the curve, the direction of the secant approaches that of the tangent at P, assuming there is just one. As a consequence, one could say that the limit
Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the behavior of a Function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely....
 of the secant's slope
Slope

Slope is used to describe the steepness, incline, gradient, or grade of a line . A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two point...
, or direction, is that of the tangent.

A chord
Chord (geometry)

A chord of a curve is a geometry line segment whose endpoints both lie on the curve.A secant or a secant line is the line extension of a chord....
 is a segment of a secant line whose both ends lie on the curve.

How the secant function is related to secant lines

Construct the unit circle
Unit circle

In mathematics, a unit circle is a circle with a 1 radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane....
 centered at the origin
Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special Point , usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space....
, and the tangent line to that unit circle at the point P = (1, 0). Draw through the origin a secant line at angle ? to the horizontal axis. For values of ? other than p/2 (90 degrees), the secant line intersects the tangent line at some point Q. Then the trigonometric secant
Trigonometric function

In mathematics, the trigonometric functions are function s of an angle. They are important in the trigonometry of Triangle and modeling Periodic function, among many other applications....
 of ? is equal to the length of the segment
Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
 of that secant line from the origin to its intersection with the tangent line at point Q.

Secant approximation

Derivative
Consider the curve defined by y = f(x) in a Cartesian coordinate system
Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
, and consider a point P with coordinates (c, f(c)) and another point Q with coordinates (c + ?x, f(c + ?x)). Then the slope
Slope

Slope is used to describe the steepness, incline, gradient, or grade of a line . A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two point...
 m of the secant line, through P and Q, is given by

The righthand side of the above equation
Equation

An equation is a mathematics Proposition, in table of mathematical symbols, that two things are exactly the same . Equations are written with an equal sign, as in...
 is a variation of Newton's
Isaac Newton

Sir Isaac Newton, Fellow of the Royal Society was an English people physicist, mathematician, Astronomy, Natural philosophy, Alchemy, and Theology and one of the the 100 in human history....
 difference quotient
Difference quotient

The primary vehicle of calculus and other higher mathematics is the Function . Its "input value" is its argument, usually a point expressible on a graph....
. As ?x approaches zero, this expression approaches the derivative
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
 of f(c), assuming a derivative exists.

There is a formula for the measure of secants.

Secant and tangent formulas for circles


If two secant lines intersect at point P, then the first secant line times the part of the first secant line outside of the circle equals the second secant line times the part of the second secant line outside of the circle.

(PA) × (PB) = (PC) × (PD)


Secant with tangent formula:

The whole secant segment times the outside segment equals the tangent squared.

(AB) × (AC) = D2


Inside secant formula:

The first part of the secant times the last side of the secant equals the other first part of the secant times the other last side of the secant.

(AB) × (BC) = (DE) × (EF)


See also

  • Differential calculus
    Calculus

    Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
  • Tangent line
    Tangent

    In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....


External links

  • With interactive applet