Tangent (Line)

A tangent line to a curve at a point is a straight line that touches the curve at that point and matches the curve's instantaneous direction (slope) there.

For a function $y = f(x)$, the tangent line at $x = a$ has equation

$y - f(a) = f'(a)(x - a),$

with slope $f'(a)$ — the derivative.

For a circle, the tangent at any point is perpendicular to the radius drawn to that point. This single fact powers many circle theorems and is the original geometric meaning of "tangent" (Latin tangere, "to touch").

Modern usage extends to:

• Tangent plane to a surface in 3D (linear approximation).
• Tangent vector to a curve in any dimension.
• Tangent space to a manifold (entire field of differential geometry).

Don't confuse the geometric tangent line with the trig tangent function $\tan\theta$ — they share the name because of an old construction relating an angle to a tangent line of the unit circle, but in modern usage they're separate concepts.