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Secant line)
In trigonometry, a secant is a particular trigonometric function, the reciprocal of the cosine function:
- sec(θ) = 1/cos(θ).
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:
- <math>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}<math>
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
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