What is the pi fraction of $\arctan (9)$? I converted it to degrees and decimal, but how is it expressed in terms of $\pi$?
$\endgroup$ 83 Answers
$\begingroup$$\frac{1}{\pi}\,\arctan(9)$ is not a rational number, so your question makes little sense.
Proof. If we assume $\frac{1}{\pi}\arctan(9)\in\mathbb{Q}$, we have that $\frac{1+9i}{\sqrt{82}}$ is a $m$-th root of unity for some $m$.
The minimal polynomial of $\frac{1+9i}{\sqrt{82}}$ over $\mathbb{Q}$ is given by $41x^4+80 x^2+41$. This is not a monic polynomial, hence it cannot be a cyclotomic polynomial. We reached a contradiction.
On the other hand, since the continued fraction of $\frac{1}{\pi}\arctan(9)$ is $$[0; 2, 6, 1, 1, 2, 16, 9, 7, 5, 1, 7, 54, 26, 9, 1, 3, 1, 2, 2,\ldots]$$ we have the approximate identity $\arctan(9)\approx \frac{1677401}{3609047}\pi$.
Now we may prove that $\frac{1}{\pi}\arctan(9)$ is a trascendental number by invoking the Gelfond-Schneider theorem. We already have that $\frac{1}{\pi i}\log\left(\frac{1+9i}{1-9i}\right)$ is an irrational number. If we assume it is an algebraic number, by GS we have that $$ (-1)^{\frac{1}{\pi i}\log\left(\frac{1+9i}{1-9i}\right)} = \frac{1+9i}{1-9i}$$ is trascendental, contradiction.
$\endgroup$ 4 $\begingroup$You can approximate it using RootApproximant
at92 = (ArcTan[9]/π // RootApproximant[#, 2] &) π
(* ((89579 + Sqrt[12802312961]) π)/436180 *)The relative error is
(at92 - ArcTan[9])/ArcTan[9] // N
(* 1.52071*10^-16 *)For a simpler, less accurate approximation
at91 = (ArcTan[9]/π // RootApproximant[#, 1] &) π
(* (1677401 π)/3609047 *)with relative error
(at91 - ArcTan[9])/ArcTan[9] // N
(* 3.04142*10^-15 *)Or using Rationalize
at9r = (ArcTan[9]/π // Rationalize[#, 10^-16] &) π
(* (85761483 π)/184521902 *)with relative error
(at9r - ArcTan[9])/ArcTan[9] // N
(* -1.52071*10^-16 *)While Rationalize[ArcTan[1]/\[Pi]] gives $1/4$, Rationalize[ArcTan[9]/\[Pi]] gives $\arctan(9)/\pi$.
`N[ArcTan[9]/\[Pi], 60]`
0.464776712522722725577810346306513918765266462161294796367476
It does not seem to be a rational number
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