- Secant line
A secant line of a
curve is a line that (locally) intersects two points on the curve. The word "secant" comes from theLatin "secare", for "to cut".It can be used to approximate the
tangent to acurve , 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'sslope , 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.ecant approximation
Consider the curve defined by "y" = "f"("x") in a
Cartesian coordinate system , and consider a point "P" withcoordinates ("c", "f"("c")) and another point "Q" with coordinates ("c" + Δ"x", "f"("c" + Δ"x")). Then theslope "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'sdifference quotient . As Δ"x" approaches zero, this expression approaches thederivative 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)
ee also
*Differential calculus
*Tangent LineExternal links
* [http://mathworld.wolfram.com/SecantLine.html Secant Line from "Mathworld"]
* [http://www.clas.ucsb.edu/staff/lee/Secant,%20Tangent,%20and%20Derivatives.htm Secant Lines, Tangent Lines, and Limit Definition of a Derivative]
* [http://mathopenref.com/secant.html Math Open Reference: Secant definition] With interactive applet
Wikimedia Foundation. 2010.