Sekante

Sekante, in mathematics, refers to two related concepts: a secant line and the trigonometric secant function. The term comes from the Latin secans, meaning “cutting,” reflecting how a secant line cuts through a curve.

Secant line: A straight line that intersects a curve at two or more points. In particular, for a circle a secant meets the circle at exactly two points, in contrast to a tangent, which touches at a single point. Secant lines are fundamental in analytic geometry for approximating curves and for defining slopes of chords.

Secant function: The secant of an angle θ is the reciprocal of the cosine, defined as sec(θ) = 1/cos(θ). The function is undefined where cos(θ) = 0 (at θ = π/2 + kπ for integers k). It is periodic with period 2π and has range (−∞, −1] ∪ [1, ∞). Examples include sec(0) = 1 and sec(π/3) = 2. The graph features vertical asymptotes at the points where cosine vanishes.

Properties and relations: The derivative of the secant function is d/dx sec(x) = sec(x) tan(x). Trigonometric identities connect secant to other functions, such as sec^2(x) = 1 + tan^2(x). Secant commonly appears in geometry, trigonometry, and calculus, often in problems involving reciprocals of cosine or in the analysis of curves and their chords.

Applications: Secant concepts are used to approximate slopes of curves via secant lines, to solve trigonometric equations, and in integration and Fourier analysis where reciprocal trigonometric functions arise. The term remains a standard part of mathematical vocabulary in several languages.