The inverse tangent is also called the arctangent and is denoted either
or arctan
. It has the
Maclaurin Series
![\begin{displaymath}
\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}](i_1016.gif) |
(1) |
A more rapidly converging form due to Euler
is given by
![\begin{displaymath}
\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}](i_1017.gif) |
(2) |
(Castellanos 1988). The inverse tangent satisfies
![\begin{displaymath}
\tan^{-1}x=-\tan^{-1}(-x)
\end{displaymath}](i_1018.gif) |
(3) |
for Positive and Negative
, and
![\begin{displaymath}
\tan^{-1}x={\textstyle{1\over 2}}\pi-\tan^{-1}\left({1\over x}\right)
\end{displaymath}](i_1019.gif) |
(4) |
for
. The inverse tangent is given in terms of other inverse trigonometric functions by
for Positive or Negative
, and
for
.
In terms of the Hypergeometric Function,
(Castellanos 1988). Castellanos (1986, 1988) also gives some curious formulas in terms of the Fibonacci Numbers,
where
and
is the largest Positive Root of
![\begin{displaymath}
8xv^4-100v^3-450xv^2+875v+625x=0.
\end{displaymath}](i_1036.gif) |
(19) |
The inverse tangent satisfies the addition Formula
![\begin{displaymath}
\tan^{-1}x+\tan^{-1}y=\tan^{-1}\left({x+y\over 1-xy}\right)
\end{displaymath}](i_1037.gif) |
(20) |
as well as the more complicated Formulas
![\begin{displaymath}
\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}](i_1038.gif) |
(21) |
![\begin{displaymath}
\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}](i_1039.gif) |
(22) |
![\begin{displaymath}
\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}](i_1040.gif) |
(23) |
the latter of which was known to Euler.
The inverse tangent Formulas are
connected with many interesting approximations to Pi
![\begin{displaymath}
\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}](i_1041.gif) |
(24) |
Euler
gave
![\begin{displaymath}
\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}](i_1042.gif) |
(25) |
where
![\begin{displaymath}
y\equiv {x^2\over 1+x^2}.
\end{displaymath}](i_1043.gif) |
(26) |
The inverse tangent has Continued Fraction representations
![\begin{displaymath}
\tan^{-1} x={\strut\displaystyle x\over\strut\displaystyle 1...
...\displaystyle 25x^2\over\strut\displaystyle 7-5x^2+\ldots}}}}.
\end{displaymath}](i_1044.gif) |
(27) |
To find
numerically, the following Arithmetic-Geometric Mean-like Algorithm can
be used. Let
Then compute
and the inverse tangent is given by
![\begin{displaymath}
\tan^{-1} x=\lim_{n\to\infty} {x\over\sqrt{1+x^2}\,a_n}
\end{displaymath}](i_1051.gif) |
(32) |
(Acton 1990).
An inverse tangent
with integral
is called reducible if it is expressible as a finite sum of the form
![\begin{displaymath}
\tan^{-1} n=\sum_{k=1} f_k \tan^{-1} n_k,
\end{displaymath}](i_1053.gif) |
(33) |
where
are Positive or Negative Integers and
are Integers
.
is reducible Iff all the Prime factors of
occur among the Prime factors of
for
, ...,
. A second Necessary and Sufficient condition is that the largest Prime factor
of
is less than
. Equivalent to the second condition is the statement that every Gregory Number
can be uniquely expressed as a sum in terms of
s for which
is a Størmer
Number (Conway and Guy 1996). To find this decomposition, write
![\begin{displaymath}
\arg(1+in)=\arg \prod_{k=1} (1+n_ki)^{f_k},
\end{displaymath}](i_1065.gif) |
(34) |
so the ratio
![\begin{displaymath}
r={\prod_{k=1} (1+n_ki)^{f_k}\over 1+in}
\end{displaymath}](i_1066.gif) |
(35) |
is a Rational Number. Equation (35) can also be written
![\begin{displaymath}
r^2(1+n^2)=\prod_{k=1} (1+{n_k}^2)^{f_k}.
\end{displaymath}](i_1067.gif) |
(36) |
Writing (33) in the form
![\begin{displaymath}
\tan^{-1} n=\sum_{k=1} f_k\tan^{-1} n_k+f\tan^{-1} 1
\end{displaymath}](i_1068.gif) |
(37) |
allows a direct conversion to a corresponding Inverse Cotangent Formula
![\begin{displaymath}
\cot^{-1} n=\sum_{k=1} f_k\cot^{-1} n_k + c\cot^{-1} 1,
\end{displaymath}](i_948.gif) |
(38) |
where
![\begin{displaymath}
c=2-f-2\sum_{k=1} f_r.
\end{displaymath}](i_1069.gif) |
(39) |
Todd (1949) gives a table of decompositions of
for
. Conway and Guy (1996) give a similar table in
terms of Størmer Numbers.
Arndt and Gosper give the remarkable inverse tangent identity
![\begin{displaymath}
\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}](i_1071.gif) |
(40) |
See also Inverse Cotangent, Tangent
References
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.''
http://www.jjj.de/hfloat/hfloatpage.html#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.
© 1996-9 Eric W. Weisstein
1999-05-26