Contents

# Contents

## Idea

In trigonometry, the tangent function$tan$” is one of the basic trigonometric functions.

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 ${\color{red}\theta}$ between that radial line and the one orthogonal to the tangent.

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

## Definition

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

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

## Properties

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

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

$\tan(i x) = i\tanh x,$

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

$\sec x + \tan x = \sum_{n \geq 0} \frac{E_n x^n}{n!}$

is the exponential generating function of a sequence of natural numbers $E_k$ that count “zigzag permutations” on the set $\{1, \ldots, k\}$, i.e., permutations $a_1 a_2 \ldots a_k$ where $a_1 \gt a_2 \lt a_3 \gt \ldots$. The tangent numbers, which are the coefficients $E_n$ when $n$ is odd, are related to Bernoulli numbers through the formula

$(-1)^{k-1} B_{2k} = \frac{2k \cdot E_{2k-1}}{2^{2k}(2^{2k} - 1)}.$

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

$2(2^{2k}-1)\zeta(2k) = \frac{E_{2k-1}\pi^{2k}}{(2k-1)!}.$

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

$\tan x = \cot x - 2\cot(2x).$

## References

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