nLab tangent function




In trigonometry, the tangent functiontantan” is one of the basic trigonometric functions.

From Mathnet

The tangent function may be understood as computing the length of the segment (shown in blue) of a tangent (whence the name) to a unit circle inside the Euclidean plane, between its point of tangency and its intersection with any radial line, as a function of the angle θ{\color{red}\theta} between that radial line and the one orthogonal to the tangent.

Since, by definition of the sine functionsinsin” and the cosine functioncoscos”, that radial line has distance sin(θ)sin(\theta) from the perpendicular line where it crosses the unit circle at cos(θ)cos(\theta) along that perpendicular line, the length of that tangential line segment equals the ratio sin(θ)/cos(θ)sin(\theta)/cos(\theta), and this is how the tangent function is often defined.


The tangent function is the ratio of the sine and the cosine, where this is defined:

θ{(k+1/2)π|k}tan(θ)sin(θ)cos(θ) \theta \,\in\, \mathbb{R} \setminus \big\{ (k+1/2)\pi \,\big\vert\, k \in \mathbb{Z} \big\} \;\;\;\;\;\;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\;\;\;\;\;\; \tan(\theta) \,\coloneqq\, \frac{\sin(\theta)}{\cos(\theta)}


  • The tangent function f(x)=tanxf(x) = \tan x is of course odd, and obeys a functional equation (‘addition formula’)

    f(x+y)=f(x)+f(y)1f(x)f(y).f(x + y) = \frac{f(x) + f(y)}{1 - f(x)f(y)}.
  • The hyperbolic analog tanhx\tanh x, related to the tangent by

    tan(ix)=itanhx,\tan(i x) = i\tanh x,

    arises in special relativity theory in connection with rapidity.

  • The MacLaurin series for the tangent function arises in enumerative combinatorics, especially when bundled with the secant function?. Here the sum

    secx+tanx= n0E nx nn!\sec x + \tan x = \sum_{n \geq 0} \frac{E_n x^n}{n!}

    is the exponential generating function of a sequence of natural numbers E kE_k that count “zigzag permutations” on the set {1,,k}\{1, \ldots, k\}, i.e., permutations a 1a 2a ka_1 a_2 \ldots a_k where a 1>a 2<a 3>a_1 \gt a_2 \lt a_3 \gt \ldots. The tangent numbers, which are the coefficients E nE_n when nn is odd, are related to Bernoulli numbers through the formula

    (1) k1B 2k=2kE 2k12 2k(2 2k1).(-1)^{k-1} B_{2k} = \frac{2k \cdot E_{2k-1}}{2^{2k}(2^{2k} - 1)}.

    They are related to values of the zeta function by the formula

    2(2 2k1)ζ(2k)=E 2k1π 2k(2k1)!.2(2^{2k}-1)\zeta(2k) = \frac{E_{2k-1}\pi^{2k}}{(2k-1)!}.

    Both of these facts may be verified by putting together corresponding facts for the cotangent function (see there) with the double angle formula

    tanx=cotx2cot(2x).\tan x = \cot x - 2\cot(2x).


Last revised on January 3, 2024 at 01:11:20. See the history of this page for a list of all contributions to it.