tangent

Tangent has multiple related meanings in geometry and trigonometry. In plane geometry, a tangent line to a curve at a given point is a line that touches the curve at that point and has the same direction as the curve there, i.e., it has first-order contact. For a circle, the tangent at a point on the circle is perpendicular to the radius drawn to that point.

In trigonometry, the tangent of an angle is defined as the ratio of the sine to the cosine: tan θ = sin θ / cos θ. The tangent function is periodic with period π, is odd tan(-θ) = -tan θ, and has vertical asymptotes at θ = π/2 + kπ. Its derivative is sec^2 θ, which equals 1 + tan^2 θ.

In calculus, the tangent line to a function y = f(x) at x0 has slope f'(x0). The line itself is y − f(x0) = f'(x0)(x − x0); the angle it makes with the x-axis satisfies tan φ = f'(x0).

In a right triangle, tan of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

Etymology and history: the term tangent comes from Latin tangere, meaning to touch; the notion of a tangent line dates to classical geometry, and the tangent function was developed in medieval and early modern trigonometry as a complement to sine and cosine.

Applications include physics, engineering, computer graphics, and navigation, where tangent functions model slopes, rates of change, and angular relationships.