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Suppose we have many expressions involving irrational ArcTan expressions (such as ArcTan[Sqrt[3],5]) along with other irrational factors (such as 1/Sqrt[2]).

I want to simplify all of these ArcTans. For example, I might want to apply N to them to get their decimal approximation, or I might want to apply Rationalize as in

N@ArcTan[Sqrt[3], 5] (* out: 1.23732 *) Rationalize[ArcTan[Sqrt[3], 5]/Pi, .0001] Pi (* out: 13 Pi/33 *)

However, I would like to be able to implement either of these options throughout large expressions involving many ArcTan functions and not act on any other forms.

Simple example would be: 

m = 1/Sqrt[3]+ Sqrt[2] ArcTan[Sqrt[3], 5]

How can I apply a rule or something to m that will allow me to implement one of these types of solutions to the ArcTan but not act on the other irrational factors?

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  • $\begingroup$ Something like m /. ArcTan[Times[x_, y_]] :> ArcTan[fff[x], ggg[y]] /; ! Element[x | y, Rationals] may work $\endgroup$ Commented Mar 12, 2014 at 21:56
  • $\begingroup$ It may, but I haven't yet been able to construct fff and ggg correctly to implement it this way. $\endgroup$ Commented Mar 13, 2014 at 14:29
  • $\begingroup$ For example m = 1/Sqrt[3] + Sqrt[2] ArcTan[Sqrt[3], 5]; f[u_] := u /. ArcTan[Times[x_, y_]] :> (Rationalize[ArcTan[Times[x, y]]/Pi, .0001] Pi) /; ! Element[x | y, Rationals]; f[m] $\endgroup$ Commented Mar 13, 2014 at 14:48

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Approaching (stealing) of Dr. Belisarius's answer which was very helpful, I present my answer:

Providing an m equation:

m = 1/Sqrt[3]+ Sqrt[2] ArcTan[Sqrt[3], 5]

$\frac{1}{\sqrt{3}}+\sqrt{2} \tan ^{-1}\left(\frac{5}{\sqrt{3}}\right)$

You can implement this type of solution:

f[u_] := u /. ArcTan[x_ y_] :> π Rationalize[ArcTan[x y]/π, 0.0001]; f[m]

$\frac{13 \pi \sqrt{2}}{33}+\frac{1}{\sqrt{3}}$

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