secant

Secant is a term used in geometry, trigonometry, and calculus. In geometry, a secant is a line that intersects a curve at two points. For a circle, a line that cuts the circle in two distinct points is called a secant; if it touches at exactly one point it is a tangent, and if it does not intersect the circle it is not a secant. The part of the secant lying inside the circle is a chord, and the segment between the intersection points is called the secant segment.

In trigonometry, the secant function, sec(x), is defined as the reciprocal of cosine: sec(x) = 1/cos(x). It is undefined where cos x = 0, at x = pi/2 + k pi. The secant function is related to other trig functions by the identity sec^2 x = 1 + tan^2 x, and by graphs with vertical asymptotes at the same points as csc and tan. Examples: sec(0) = 1; sec(pi/3) = 2.

In calculus and numerical methods, a secant line to a function y = f(x) through points (x1, f(x1)) and (x2, f(x2)) has slope m = [f(x2) - f(x1)] / (x2 - x1). As x2 approaches x1, the secant line tends to the tangent line with slope f'(x). The secant method is an iterative root-finding algorithm that uses secant lines to approximate a root of f.

Etymology: Secant comes from Latin secans, present participle of secare "to cut," reflecting its role in intersecting curves as if cutting through them.