Tangent (mathematics and geometry)

Tangent in trigonometry

For an angle measured in a right triangle, the tangent is the ratio of the length of the side opposite the angle to the length of the adjacent side. On the unit circle it is the ratio sin θ / cos θ. The tangent function is periodic with period π, is odd (tan(−θ)=−tan θ), and has vertical asymptotes where cos θ = 0.

Tangent lines and calculus

In geometry a tangent line touches a curve at a single point and matches the curve's instantaneous direction there. Analytically, the slope of a tangent line at a point is the limit of slopes of secant lines and, when the curve is differentiable, equals the derivative. The tangent line gives the first-order linear approximation f(x) ≈ f(a)+f'(a)(x−a).

Tangent planes and tangent spaces

Higher-dimensional analogues include tangent planes to surfaces in three-space and tangent spaces to differentiable manifolds. A tangent plane approximates a surface near a point by a flat affine plane. The tangent space at a point on a manifold is the vector space of possible velocity vectors of curves through that point and underlies much of differential geometry.

Key identities and facts

• Definition: tan θ = sin θ / cos θ.
• Derivative: d/dx[tan x] = sec^2 x = 1 + tan^2 x.
• Addition formula: tan(a + b) = (tan a + tan b)/(1 − tan a tan b) when denominator ≠ 0.
• Poles: tan x has simple poles at x = π/2 + kπ for integer k.

History, usage and distinctions

The word derives from Latin tangens, 'touching'. Ideas of tangency go back to classical geometry; the calculus-era formalization of tangents and instantaneous slope emerged with 17th-century advances. In common language a tangent may mean a remark that merely 'touches' the main topic. Mathematically, tangents contrast with secants (which cut a curve) and normals (which are perpendicular to tangents).