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Secant line
 
 
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A secant line of a curveCurve

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and con...
 is a line that (locally) intersects two pointPoint (geometry)

A spatial point is an entity with a location in space but no extent....
s on the curve. The word secant comes from the LatinLatin

Latin is an ancient Indo-European language originally spoken in Latium, the region immediately surrounding Rome....
 secare, for to cut.

It can be used to approximate the tangentTangent

In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonomet...
 to a curveCurve

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and con...
, at some point P. If the secant to a curve is defined by two pointPoint (geometry)

A spatial point is an entity with a location in space but no extent....
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 limitLimit (mathematics)

In mathematics, the concept of a "limit" is used to describe the behavior of a function as its argument either gets "close" ...
 of the secant's slopeSlope

The slope or the gradient is commonly used to describe the measurement of the steepness, incline or grade of a straigh...
, or direction, is that of the tangent.

A chordChord (geometry)

A chord of a curve is a geometric line segment whose endpoints both lie on the curve....
 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 circleUnit circle Summary

In mathematics, a unit circle is a circle with unit radius, i.e., a circle whose radius is 1....
 centered at the originOrigin (mathematics)

In mathematics, the origin of a coordinate system is the point where the axes of the system intersect....
, 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 secantTrigonometric function

In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeli...
 of ? is equal to the length of the segmentLine segment

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

Secant approximation


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

In mathematics, the Cartesian coordinate system is used to uniquely determine each point in the plane through two numbers, u...
, and consider a point P with coordinates  (c, f(c)) and another point Q with coordinates (c + ?x, f(c + ?x)). Then the slopeSlope

The slope or the gradient is commonly used to describe the measurement of the steepness, incline or grade of a straigh...
 m of the secant line, through P and Q, is given by

The righthand side of the above equationEquation

An equation is a mathematical statement, in symbols, that two things are the same....
 is a variation of Newton'sIsaac Newton

[[[Old Style and New Style dates|OS]]: [[25 December]] [[1642]] [[20 March]] [[1727]]] was an [[England|English]] [[physics|physicist,]]...
 difference quotientDifference quotient

The primary vehicle of calculus and other higher mathematics is the function....
. As ?x approaches zero, this expression approaches the derivativeFacts About Derivative

In mathematics, the derivative is defined as the instantaneous rate of change of a function....
 of f(c), assuming a derivative exists.

Secant and Tangent Formulas for circles


the first segment to the point on a circle times the whole segment equals the first segment to the other point on a circle times the other whole segment.

(AB)x(AC)=(DE)x(DF)

Secant with Tangent Formula:

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

(AB)x(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 and the other last side of the secant.

(AB)x(BC)=(DE)x(EF)

See also

  • Differential calculusCalculus

    Calculus is a central branch of mathematics, developed from algebra and geometry....
  • Tangent LineTangent

    In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonomet...


External links

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