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.
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.
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 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.