Tangent

A tangent is a straight line that touches a curve at exactly one point, with its direction at that point matching the direction of the curve at that point. The word comes from the Latin "tangens", meaning "touching".

The concept of the tangent is foundational to both geometry and calculus. When you want to understand how a curve changes at a particular point, the tangent line at that point shows the linear change closest to the curve in the neighbourhood of the point.

Tangent to a circle

The simplest case is a circle. The tangent line to a circle touches the circle at exactly one point and is perpendicular (at 90 degrees) when compared to the radius at the point of tangency. If a line touches a circle at two points, it is called a secant, not a tangent. The distance from the centre of the circle to the tangent line equals the length of the radius.

Tangent and the derivative

In calculus, the slope of the tangent line to the graph of a function f at the point x equals the derivative f'(x). The tangent line can be written as y = f(a) + f'(a)(x − a). This expresses the idea that the tangent is found by taking the limit of the slope of secant lines as the two points approach each other.

Special cases

Not every curve has a tangent at every point. At a point where the curve has a break or corner (such as the graph of |x| at zero), the derivative does not exist, and thus the tangent is undefined. There also exist vertical tangents whose slope is undefined. In some cases, the tangent line crosses the curve at the point of tangency, as at an inflection point.

The term in trigonometry

The word "tangent" is also the name of one of the trigonometric functions, tan(θ), defined as the ratio of sine to cosine. The geometric connection arose from the length of a line segment tangent to a unit circle. Although both uses are historically linked, their mathematical meanings are distinct.