[eriksank]

I think it may be possible to phrase a claim for this.

for any almostZero, it is possible to find an integer m1, so that the m1.2.pi + pi/2 is greater than 1/almostZero and therefore for which sin(m1.2.pi+pi/2) =1; 1/(m1.2.pi + pi/2) will be smaller than 1/almostZero. It is also possible to find an integer m2, so that m2.pi+pi is greater than 1/almostZero and therefore for which sin(m2.pi+pi) =0; 1/(m2.pi+pi) will be smaller than 1/almostZero. The claim: find me an almostZero for it is not possible to find such m1,m2.

This could be a possibility, but I haven't written the program yet ;-)