Tangents

Tangents are a fundamental concept in geometry and analysis with several related meanings. In geometry, a tangent to a curve at a point is a line that touches the curve at that point and, for a smooth curve, has the same direction as the curve there. It can be understood as the limit of secant lines through the point as the second point on the curve approaches the point. In calculus, the tangent line provides the best linear approximation to the curve at that point; its slope equals the derivative of the curve at that point.

For a circle, the tangent at a point is perpendicular to the radius drawn to that point. This perpendicularity underpins many geometric constructions and proofs and helps distinguish tangents from secants.

The tangent function, denoted tan(x), is a trigonometric function defined as tan(x) = sin(x)/cos(x). It is defined wherever cos(x) ≠ 0, and it has vertical asymptotes at x = π/2 + kπ (k an integer). The tangent function is odd and has period π. It arises naturally when considering angles and right triangles, and it is related to the slope of lines forming angle x with the x-axis.

Uses and properties of tangents include linear approximation in calculus, small-angle approximations such as tan(x) ≈ x for small x, and various trigonometric identities. The term derives from the Latin tangens, meaning “touching,” reflecting the geometric idea of a line that touches a curve or circle without crossing it at the point of contact.