Secant

In trigonometry, a secant is a particular trigonometric function, the reciprocal of the cosine function.

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A secant line of a curve is that line which intersects two (or more) points upon the curve. Note that this use of "secant" comes from the Latin "secare", for "to cut"; this is not a reference to the trigonometric function.

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.

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 = Δy / Δx = [f(c + Δx) - f(c)] / [(c + Δx) - c] = [f(c + Δx) - f(c)] / Δx

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