Secant Guide, Meaning , Facts, Information and Description

A secant line of a curve is that line which intersects two (or more) points upon the curve. The word secant comes from the Latin secare, for to cut.

It can be used to approximate the tangent to a curve, at some point P. If the secant to a curve is defined by two points, 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 of the secant's slope, or direction, is that of the tangent.

Table of contents

1 How the trigonometric secant function is related to this concept 2 Secant approximation 3 See also

How the trigonometric secant function is related to this concept

The absolute value of the secant function of trigonometry, which is the reciprocal of the cosine function, is the length of a segment of a secant line to the unit circle centered at the origin in the Cartesian plane, running from the origin to the tangent line x = 1. If the segment passes through the point (cos θ, sin θ), then the trigonometric secant of θ is positive; if it passes through the antipodal point, then the secant of θ is negative.

Secant approximation

Consider the curve defined by y = f(x) in a Cartesian coordinate system, and consider a point P with coordinates (c, f(c)) and another point Q with coordinates (c + Δx, f(c + Δx)). Then the slope m of the secant line, through P and Q, is given by:

The righthand side of the above equation is a variation of Newton's difference quotient. As Δx approaches zero, this expression approaches the derivative of f(c), assuming a derivative exists.

See also

derivative, differential calculus

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