# Secant

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*+ Δ

*x*,

*f*(

*c*+ Δ

*x*)). Then the slope

*m*of the secant line, through

*P*and

*Q*, is given by:

*x*approaches zero, this expression approaches the derivative of

*f*(

*c*), assuming a derivative exists.

See also: derivative, differential calculus