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Inverse Tangent


\begin{figure}\begin{center}\BoxedEPSF{ArcTanReIm.epsf scaled 700}\end{center}\end{figure}

The inverse tangent is also called the arctangent and is denoted either $\tan^{-1}x$ or arctan $x$. It has the Maclaurin Series

\tan^{-1} x=\sum_{n=0}^\infty {(-1)^nx^{2n+1}\over 2n+1} = x...
...+{\textstyle{1\over 5}} x^5-{\textstyle{1\over 7}} x^7+\ldots.
\end{displaymath} (1)

A more rapidly converging form due to Euler is given by
\tan^{-1} x=\sum_{n=0}^\infty {2^{2n}(n!)^2\over(2n+1)!} {x^{2n+1}\over(1+x^2)^{n+1}}
\end{displaymath} (2)

(Castellanos 1988). The inverse tangent satisfies
\end{displaymath} (3)

for Positive and Negative $x$, and
\tan^{-1}x={\textstyle{1\over 2}}\pi-\tan^{-1}\left({1\over x}\right)
\end{displaymath} (4)

for $x\geq 0$. The inverse tangent is given in terms of other inverse trigonometric functions by
$\displaystyle \tan^{-1}x$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\pi-\cos^{-1}\left({x\over\sqrt{x^2+1}}\right)$ (5)
  $\textstyle =$ $\displaystyle \cot^{-1}(-x)-{\textstyle{1\over 2}}\pi={\textstyle{1\over 2}}\pi-\cot^{-1} x$ (6)
  $\textstyle =$ $\displaystyle \sin^{-1}\left({x\over\sqrt{x^2+1}}\right)$ (7)

for Positive or Negative $x$, and
$\displaystyle \tan^{-1}x$ $\textstyle =$ $\displaystyle \cos^{-1}\left({1\over\sqrt{x^2+1}}\right)$ (8)
  $\textstyle =$ $\displaystyle \cot^{-1}\left({1\over x}\right)$ (9)
  $\textstyle =$ $\displaystyle \csc^{-1}\left({\sqrt{x^2+1}\over x}\right)$ (10)
  $\textstyle =$ $\displaystyle \sec^{-1}(\sqrt{x^2+1}\,)$ (11)

for $x\geq 0$.

In terms of the Hypergeometric Function,

$\displaystyle \tan^{-1}x$ $\textstyle =$ $\displaystyle x \,{}_2F_1(1, {\textstyle{1\over 2}}; {\textstyle{3\over 2}}; -x^2)$ (12)
  $\textstyle =$ $\displaystyle {x\over 1+x^2}\, {}_2F_1\left({1, 1; {\textstyle{3\over 2}}; {x^2\over 1+x^2}}\right)$ (13)

(Castellanos 1988). Castellanos (1986, 1988) also gives some curious formulas in terms of the Fibonacci Numbers,
$\displaystyle \tan^{-1}x$ $\textstyle =$ $\displaystyle \sum_{n=0}^\infty {(-1)^n F_{2n+1}t^{2n+1}\over 5^n(2n+1)}$ (14)
  $\textstyle =$ $\displaystyle 5\sum_{n=0}^\infty {(-1)^n{F_{2n+1}}^2\over (2n+1)(u+\sqrt{u^2+1}\,)^{2n+1}}$ (15)
  $\textstyle =$ $\displaystyle \sum_{n=0}^\infty {(-1)^n5^{n+2}{F_{2n+1}}^3\over(2n+1)(v+\sqrt{v^2+5}\,)^{2n+1}},$ (16)

$\displaystyle t$ $\textstyle \equiv$ $\displaystyle {2x\over 1+\sqrt{4x^2\over 5}}$ (17)
$\displaystyle u$ $\textstyle \equiv$ $\displaystyle {5\over 4x} \left({1+\sqrt{1+{\textstyle{24\over 25}}x^2}\,}\right),$ (18)

and $v$ is the largest Positive Root of
\end{displaymath} (19)

The inverse tangent satisfies the addition Formula

\tan^{-1}x+\tan^{-1}y=\tan^{-1}\left({x+y\over 1-xy}\right)
\end{displaymath} (20)

as well as the more complicated Formulas
\tan^{-1}\left({1\over a-b}\right)=\tan^{-1}\left({1\over a}\right)+\tan^{-1}\left({b\over a^2-ab+1}\right)
\end{displaymath} (21)

\tan^{-1}\left({1\over a}\right)=2\tan^{-1}\left({1\over 2a}\right)-\tan^{-1}\left({1\over 4a^3+3a}\right)
\end{displaymath} (22)

\tan^{-1}\left({1\over p}\right)=\tan^{-1}\left({1\over p+q}\right)+\tan^{-1}\left({q\over p^2+pq+1}\right),
\end{displaymath} (23)

the latter of which was known to Euler. The inverse tangent Formulas are connected with many interesting approximations to Pi

\tan^{-1}(1+x) = {\textstyle{1\over 4}}\pi + {\textstyle{1\o...
...extstyle{1\over 48}}x^6+ {\textstyle{1\over 112}}x^7 + \ldots.
\end{displaymath} (24)

Euler gave
\tan^{-1} x = {y\over x}\left({{2\over 3} y+{2\cdot 4\over 3...
...y^2+{2\cdot 4\cdot 6\over 3\cdot 5\cdot 7}y^3 +\ldots}\right),
\end{displaymath} (25)

y\equiv {x^2\over 1+x^2}.
\end{displaymath} (26)

The inverse tangent has Continued Fraction representations

\tan^{-1} x={\strut\displaystyle x\over\strut\displaystyle 1...
...\displaystyle 25x^2\over\strut\displaystyle 7-5x^2+\ldots}}}}.
\end{displaymath} (27)

To find $\tan^{-1}x$ numerically, the following Arithmetic-Geometric Mean-like Algorithm can be used. Let
$\displaystyle a_0$ $\textstyle =$ $\displaystyle (1+x^2)^{-1/2}$ (28)
$\displaystyle b_0$ $\textstyle =$ $\displaystyle 1.$ (29)

Then compute
$\displaystyle a_{i+1}$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(a_i+b_i)$ (30)
$\displaystyle b_{i+1}$ $\textstyle =$ $\displaystyle \sqrt{a_{i+1}b_i},$ (31)

and the inverse tangent is given by
\tan^{-1} x=\lim_{n\to\infty} {x\over\sqrt{1+x^2}\,a_n}
\end{displaymath} (32)

(Acton 1990).

An inverse tangent $\tan^{-1} n$ with integral $n$ is called reducible if it is expressible as a finite sum of the form

\tan^{-1} n=\sum_{k=1} f_k \tan^{-1} n_k,
\end{displaymath} (33)

where $f_k$ are Positive or Negative Integers and $n_i$ are Integers $<n$. $\tan^{-1} m$ is reducible Iff all the Prime factors of $1+m^2$ occur among the Prime factors of $1+n^2$ for $n=1$, ..., $m-1$. A second Necessary and Sufficient condition is that the largest Prime factor of $1+m^2$ is less than $2m$. Equivalent to the second condition is the statement that every Gregory Number $t_x=\cot^{-1}x$ can be uniquely expressed as a sum in terms of $t_m$s for which $m$ is a Størmer Number (Conway and Guy 1996). To find this decomposition, write
\arg(1+in)=\arg \prod_{k=1} (1+n_ki)^{f_k},
\end{displaymath} (34)

so the ratio
r={\prod_{k=1} (1+n_ki)^{f_k}\over 1+in}
\end{displaymath} (35)

is a Rational Number. Equation (35) can also be written
r^2(1+n^2)=\prod_{k=1} (1+{n_k}^2)^{f_k}.
\end{displaymath} (36)

Writing (33) in the form
\tan^{-1} n=\sum_{k=1} f_k\tan^{-1} n_k+f\tan^{-1} 1
\end{displaymath} (37)

allows a direct conversion to a corresponding Inverse Cotangent Formula
\cot^{-1} n=\sum_{k=1} f_k\cot^{-1} n_k + c\cot^{-1} 1,
\end{displaymath} (38)

c=2-f-2\sum_{k=1} f_r.
\end{displaymath} (39)

Todd (1949) gives a table of decompositions of $\tan^{-1} n$ for $n\leq 342$. Conway and Guy (1996) give a similar table in terms of Størmer Numbers.

Arndt and Gosper give the remarkable inverse tangent identity

\sin\left({\sum_{k=1}^{2n+1} \tan^{-1} a_k}\right)= {(-1)^n\...
...1}\right)}\right]\over \sqrt{\prod_{j=1}^{2n+1} ({a_j}^2+1)}}.
\end{displaymath} (40)

See also Inverse Cotangent, Tangent


Abramowitz, M. and Stegun, C. A. (Eds.). ``Inverse Circular Functions.'' §4.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79-83, 1972.

Acton, F. S. ``The Arctangent.'' In Numerical Methods that Work, upd. and rev. Washington, DC: Math. Assoc. Amer., pp. 6-10, 1990.

Arndt, J. ``Completely Useless Formulas.''

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Item 137, Feb. 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142-143, 1987.

Castellanos, D. ``Rapidly Converging Expansions with Fibonacci Coefficients.'' Fib. Quart. 24, 70-82, 1986.

Castellanos, D. ``The Ubiquitous Pi. Part I.'' Math. Mag. 61, 67-98, 1988.

Conway, J. H. and Guy, R. K. ``Størmer's Numbers.'' The Book of Numbers. New York: Springer-Verlag, pp. 245-248, 1996.

Todd, J. ``A Problem on Arc Tangent Relations.'' Amer. Math. Monthly 56, 517-528, 1949.

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© 1996-9 Eric W. Weisstein