Definitions

# Secant line

A secant line of a curve is a line that (locally) intersects two points on 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.

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 centered at the origin, and the tangent line to that unit circle at the point P = (1, 0). Draw through the origin a secant line at angle θ to the horizontal axis. For values of θ other than π/2 (90 degrees), the secant line intersects the tangent line at some point Q. Then the trigonometric secant of θ is equal to the length of the segment 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 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

$m = frac\left\{Delta y\right\}\left\{Delta x\right\} = frac\left\{f\left(c + Delta x\right) - f\left(c\right)\right\}\left\{\left(c + Delta x\right) - c\right\} = frac\left\{f\left(c + Delta x\right) - f\left(c\right)\right\}\left\{Delta x\right\}.$

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.

## 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)