Tangent

Tangent is a term with distinct meanings in mathematics and in general language, each sharing the core concept of touching at a single point or line.

In Mathematics:

In geometry, a tangent is a line that touches a curve or surface at a single point, called the point of tangency, without crossing over at that point. Formally, a tangent line is the limit of a secant line as two points of intersection between the secant line and the curve approach each other.

• Tangent to a Circle: A line is tangent to a circle if and only if it intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at the point of tangency.

• Tangent to a Curve: For a more general curve, the tangent line at a point is the line that "best approximates" the curve at that point. This is formally defined using the concept of a derivative in calculus. The slope of the tangent line at a point on a curve represented by a function f(x) is given by the derivative f'(x) evaluated at that point.

• Tangent Plane: The concept extends to three dimensions, where a tangent plane touches a surface at a single point.

In trigonometry, tangent (often abbreviated as tan) refers to a trigonometric function. In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle. Formally, tan(θ) = opposite/adjacent. The tangent function is periodic with period π.

In General Language:

The word "tangent" is also used metaphorically to describe a sudden digression or deviation from a main topic or line of thought. To "go off on a tangent" means to veer away from the primary subject and discuss something related but ultimately distinct. This usage retains the sense of touching the main subject at a single point before diverting away.