Why is it that if we are picking dots out of sine or cosine using the formula sin(n) or cos(n), we never get any pattern?

This behaviour is caused by the fact that the distance between the dots is an integer, whereas the length of the period is an irrational number. If the distance and the period were commeasurable, precisely, if their ratio was a rational number, then we would get periodicity.

As a simple example of this phenomenon we take the sequence {cos(nπ/2)}. Now the distance between the dots is π/2, which is commeasurable with the period 2π (their mutual ratio is 4 or 1/4, a rational number in any case).

We see that now there is a periodicity, in fact the sequence goes {1, 0, −1, 0, 1, 0, −1,...}. By the way, this sequence preserves the properties of cosine: It is bounded, not monotonne, and diverges.

If we try {cos(2nπ)}, then we again get periodicity in the sequence as the mutual ratio of dot-distance and periodicity of cosine is 1, a rational number. This time the sequence goes {1, 1, 1, 1,...}, it is monotone and converges to 1. Thus it is a nice example of how properties of the function can be "improved" by smartly chosing the right points from its graph.

Conclusion: If the ratio of the period of sine/cosine and the step at which the points are taken is a rational number, then the resulting sequence is periodic and might be very nice. If this ratio is irrational (for instance sequences like {sin(n)}, {sin(−3n)}, {cos(6n)} etc.), then the resulting sequence is not periodic, not monotone, does not have a limit and keeps oscillating in the largest possible way, that is, it keeps reaching as close as we want to −1 and 1.