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
, with P
fixed and Q
variable, as Q
along the curve, the direction of the secant approaches that of the tangent
(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.
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 righthand side, of the above equation
, is a variation of Newton's difference quotient
. As Δx
approaches zero, this expression approaches the derivative
), assuming a derivative exists.
See also: derivative, differential calculus