Tangent

$\begin{figure}\begin{center}\BoxedEPSF{Tangent.epsf}\end{center}\end{figure}$

$\begin{figure}\begin{center}\BoxedEPSF{TangentReIm.epsf scaled 680}\end{center}\end{figure}$

The tangent function is defined by

$\begin{displaymath} \tan x\equiv{\sin x\over\cos x}, \end{displaymath}$

(1)

where $\sin x$ is the Sine function and $\cos x$ is the Cosine function. The word ``tangent,'' however, also has an important related meaning as a Line or Plane which touches a given curve or solid at a single point. These geometrical objects are then called a Tangent Line or Tangent Plane, respectively.

The Maclaurin Series for the tangent function is

$\displaystyle \tan x$	$\textstyle =$	$\displaystyle \sum_{n=0}^\infty {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}\over (2n)!} x^{2n-1} + \ldots$
	$\textstyle =$	$\displaystyle x + {\textstyle{1\over 3}}x^3 + {\textstyle{2\over 15}}x^5 + {\textstyle{17\over 315}}x^7 + {\textstyle{62\over 2835}}x^9 + \ldots,$	(2)

where

is a Bernoulli Number.

$\tan x$ is Irrational for any Rational $x\not =0$ , which can be proved by writing $\tan x$ as a Continued Fraction

$\begin{displaymath} \tan x={\strut\displaystyle x\over{\strut\displaystyle 1-{x^... ...displaystyle 5-{x^2\over{\strut\displaystyle 7-\ldots}}}}}}}}. \end{displaymath}$

(3)

Lambert derived another Continued Fraction expression for the tangent,

$\displaystyle \tan x$

$\textstyle =$

$\displaystyle {\strut\displaystyle 1\over {\strut\displaystyle 1\over\strut\dis... ...isplaystyle 1\over{\strut\displaystyle 7\over\strut\displaystyle x}-\ldots}}}}.$

(4)

An interesting identity involving the Product of tangents is

$\begin{displaymath} \prod_{k=1}^{\lfloor (n-1)/2\rfloor} \tan\left({k\pi\over n}... ...t)=\cases{ \sqrt{n} & for $n$\ odd\cr 1 & for $n$\ even,\cr} \end{displaymath}$

(5)

where $\left\lfloor{x}\right\rfloor$ is the Floor Function. Another tangent identity is

$\begin{displaymath} \tan(n\tan^{-1} x)={1\over i}{(1+ix)^n-(1-ix)^n\over (1+ix)^n+(1-ix)^m} \end{displaymath}$

(6)

(Beeler et al. 1972, Item 16).

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Circular Functions.'' §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 71-79, 1972.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Spanier, J. and Oldham, K. B. ``The Tangent $\tan(x)$ and Cotangent $\cot(x)$ Functions.'' Ch. 34 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 319-330, 1987.