So it is not clear enough for users what the command will do but the suer expects?

NB: I remember something 'similar' when ordering of variables matters, but VectorCalculus goes its own route. Personally I would prefer (in this case) to have at hand an explicit curve (to check before) + formula, though commands would be longer and would have to be combined.

Actually the package does something towards that with the 'inert' option:

please post as text, not as picture - so one can copy it and may see possible typos

One typo is: Pi, not pi (upper case !)

Why you want that and for what (as you recognize in you reply to Kitonum: that may be very time consuming due to magnitude of the symmetric group)?

Edited: The most trivial first test may be: the entries give the same set. Concerning their 'rank' I am not that sure in the square case one may read it as endomophism up to isomorphisms, so this should do as well as first check. A kind of Rubik's cube?

I like Carl's way, so I continue here.

Note that |theta_n| <= 1.6, so for large n it does not contribute much
to the expression Pi*n-theta_n. Setting it to 0 and feeding your data
that directly should give a good guess by tan(theta_n) = float.

For the original parameters that gives a guess with 2 leading decimals
at n=100 and with 5 decimals for n=1000 (providing correct magnitude)
i.e. relative error is ~ 1e-2 resp. 1e-5

Yes, it can be seen as Fourier integral (or as Fourier-Cosine integral), taking the real part of it.

http://de.wikipedia.org/wiki/Monte-Carlo-Algorithmus#Probabilistische_Bestimmung_der_Zahl_Pi

Instead of using the unitcircle and its according square like in the link
you use the first quadrant and its square. By symmetry it is fine as well.

Edited: crawling in dusty memories "Box-Muller" comes up as buzz word

Have not tried the "image code", but copied your first one, it gives the same (after re-writing, using evalc and  the assumption)

I think, that actually it does not depend on sign(x), and x need not be real.

Edited: acer was faster. As usual. Anyway: try x = I ( = sqrt(-1) )

well ... actually I find it "astonishing" to advice to 'turn off protection' ...

sic.                       

@cskoog Nice, really.

If this is teaching Maple or how to use it - ok. Else that approach produces mathematical illiterates. And I assume the latter - else the question might have been "how can one get ... as result?"

I understand that in most cases - and at least initially - the question
may be about real functions. Even there it is unclear, even if one uses
continous/differentiable/analytic functions only. Finding the image one
will often needs to find maximum and minimum - having already the maximal
source at hand (where already the 'discont' command does not give all
needed information - in some cases only by Numerics).

A brute example: define a function through a series, like 2F1 in the open
unit circle. All CAS will extend to a cut plane. But by compex analysis.

I could imagine to provide such, but for 'scholarly' situations only.

PS: in English it is ~ a germ finds its (maximal) domain of holomorphy,
it was how the ancients thought about Riemannian surfaces (and beyond).

For example you 'un-comment' "if convert ..." by a "#" and use the following instead

t:= {A,B,C,d,E,G,i,F,H,J,K,L}:
if min(t)=3 and max(t)=14 and 10 <= nops(t) then
  n:=n+1: 
  M[n]:=t:  
end if;

If you then display M:=convert(M, set); (semikolon !) you will see solutions
in the range. But there are only 10 solutions (so not all different) in each
of those cases.

Hope you understand how to proceed to your desired goal (even if you do
not need to understand the 'remove' command).

PS: I do not think that your Prof expects an exhaustive search, but I do not
have a better idea (at least for short).

These are not functions.

For the first I would have replied: Maple works over the Complex - but
sqrt is not a function there, however one (by choice) can extend it into
the negative axis.

For the second: similar, but different.

Ok, I understood the question. But doubt a bit in satisfactory answers.
Which does not prevent help by a CAS.

PS: once I learned "ein Keim findet selbst sein Existenzgebiet".

It says: you just can return a set.

If sampling from a loop do something like s:={} and in the loop s:= s union {currentResult}
Edited: For loops see the better solutions from the answers by acer and Carl Love