Inverse Tangent

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

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

A more rapidly converging form due to Euler is given by

$$\tan^{-1} x=\sum_{n=0}^\infty {2^{2n}(n!)^2\over(2n+1)!} {x^{2n+1}\over(1+x^2)^{n+1}}$$ (2)

(Castellanos 1988). The inverse tangent satisfies

$$\tan^{-1}x=-\tan^{-1}(-x)$$ (3)

for Positive and Negative $x$, and

$$\tan^{-1}x={\textstyle{1\over 2}}\pi-\tan^{-1}\left({1\over x}\right)$$ (4)

for $x\geq 0$. The inverse tangent is given in terms of other inverse trigonometric functions by

$$\tan^{-1}x = {\textstyle{1\over 2}}\pi-\cos^{-1}\left({x\over\sqrt{x^2+1}}\right)$$ (5)

$$= \cot^{-1}(-x)-{\textstyle{1\over 2}}\pi={\textstyle{1\over 2}}\pi-\cot^{-1} x$$ (6)

$$= \sin^{-1}\left({x\over\sqrt{x^2+1}}\right)$$ (7)

for Positive or Negative $x$, and

$$\tan^{-1}x = \cos^{-1}\left({1\over\sqrt{x^2+1}}\right)$$ (8)

$$= \cot^{-1}\left({1\over x}\right)$$ (9)

$$= \csc^{-1}\left({\sqrt{x^2+1}\over x}\right)$$ (10)

$$= \sec^{-1}(\sqrt{x^2+1}\,)$$ (11)

for $x\geq 0$.

In terms of the Hypergeometric Function,

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

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

(Castellanos 1988). Castellanos (1986, 1988) also gives some curious formulas in terms of the Fibonacci Numbers,

$$\tan^{-1}x = \sum_{n=0}^\infty {(-1)^n F_{2n+1}t^{2n+1}\over 5^n(2n+1)}$$ (14)

$$= 5\sum_{n=0}^\infty {(-1)^n{F_{2n+1}}^2\over (2n+1)(u+\sqrt{u^2+1}\,)^{2n+1}}$$ (15)

$$= \sum_{n=0}^\infty {(-1)^n5^{n+2}{F_{2n+1}}^3\over(2n+1)(v+\sqrt{v^2+5}\,)^{2n+1}},$$ (16)

where

$$t \equiv {2x\over 1+\sqrt{4x^2\over 5}}$$ (17)

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

and $v$ is the largest Positive Root of

$$8xv^4-100v^3-450xv^2+875v+625x=0.$$ (19)

The inverse tangent satisfies the addition Formula

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

as well as the more complicated Formulas

$$\tan^{-1}\left({1\over a-b}\right)=\tan^{-1}\left({1\over a}\right)+\tan^{-1}\left({b\over a^2-ab+1}\right)$$ (21)

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

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

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

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

Euler gave

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

where

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

The inverse tangent has Continued Fraction representations

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

To find $\tan^{-1}x$ numerically, the following Arithmetic-Geometric Mean-like Algorithm can be used. Let

$$a_0 = (1+x^2)^{-1/2}$$ (28)

$$b_0 = 1.$$ (29)

Then compute

$$a_{i+1} = {\textstyle{1\over 2}}(a_i+b_i)$$ (30)

$$b_{i+1} = \sqrt{a_{i+1}b_i},$$ (31)

and the inverse tangent is given by

$$\tan^{-1} x=\lim_{n\to\infty} {x\over\sqrt{1+x^2}\,a_n}$$ (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

$$\tan^{-1} n=\sum_{k=1} f_k \tan^{-1} n_k,$$ (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

$$\arg(1+in)=\arg \prod_{k=1} (1+n_ki)^{f_k},$$ (34)

so the ratio

$$r={\prod_{k=1} (1+n_ki)^{f_k}\over 1+in}$$ (35)

is a Rational Number. Equation (35) can also be written

$$r^2(1+n^2)=\prod_{k=1} (1+{n_k}^2)^{f_k}.$$ (36)

Writing (33) in the form

$$\tan^{-1} n=\sum_{k=1} f_k\tan^{-1} n_k+f\tan^{-1} 1$$ (37)

allows a direct conversion to a corresponding Inverse Cotangent Formula

$$\cot^{-1} n=\sum_{k=1} f_k\cot^{-1} n_k + c\cot^{-1} 1,$$ (38)

where

$$c=2-f-2\sum_{k=1} f_r.$$ (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

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