secantes

Secantes, or secants, are concepts used in geometry and trigonometry. In geometry, a secant is a line that intersects a circle in two distinct points. The term comes from the Latin secans, meaning “cutting.” In trigonometry, the secant is the reciprocal of the cosine function, defined for any angle x by sec(x) = 1/cos(x).

Geometrically, a secant line crosses a circle in two points, and the portion of the line between those intersections is called a secant segment. If the line passes through the circle’s center, it is a diameter, which is a special case of a secant with maximum possible length inside the circle. Secants are related to other circle concepts such as chords, tangents, and the power of a point, which describes relationships between external points, secant segments, and tangent lengths.

In trigonometry, the secant function is defined on the unit circle as sec(θ) = 1/cos(θ). It is undefined where cos(θ) = 0, at θ = π/2 + kπ for any integer k. The graph of secant has vertical asymptotes at those angles and a period of 2π; it takes values with absolute value at least 1. The derivative is d/dθ sec(θ) = sec(θ) tan(θ), and an antiderivative is ∫ sec(θ) dθ = ln|sec(θ) + tan(θ)| + C.

Applications of secants span geometry, trigonometry, and calculus, including solving triangles, modeling circular motion, and analyzing periodic phenomena. The term is used in multiple languages with similar roots, reflecting the shared idea of a line or function that relates to cutting or dividing a circle.