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{Delta y}{Delta x} = frac{f(c + Delta x) - f(c)}{(c + Delta x) - c} = frac{f(c + Delta x) - f(c)}{Delta x}.

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.


Secant with Tangent Formula:

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


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.


See also

External links

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