tangenttien

Tangenttien is the Finnish genitive plural form of tangent and, in mathematics, refers to tangents in various contexts, most commonly tangent lines to curves and the tangent function in trigonometry. The term can denote both the geometric notion of touching a curve and the trigonometric function, as well as tangents to circles.

In geometry, a tangent line to a curve at a point P is a straight line that shares the same slope as the curve at P and passes through P. It serves as the best linear approximation to the curve near P. If the curve is differentiable at P, the tangent line is unique, and its equation can be written as y − y0 = f′(x0)(x − x0) for a function y = f(x).

For a circle of radius r centered at (h, k), a tangent line at the point of tangency is perpendicular to the radius drawn to that point. The distance from the circle’s center to the tangent line equals r.

The tangent function, tan, is the ratio of sine to cosine (tan x = sin x / cos x). It has period π and vertical asymptotes where cos x = 0. Its derivative is sec²x, and it admits a unit circle interpretation linking tan to the slope of lines through the origin with angle x.

Applications of tangent concepts include linear approximation, optimization, and physics (for example, instantaneous velocity as a tangent slope) as well as computer graphics and engineering.