Since no integer $N$ is a rational multiple of $\pi$ it's obvious that $\sin N$ and $\cos N$ will not give any 'nice' values for any $N$. Actually, I thought the values would get essentially random for large $N$.
But it's not the case. If one plots the logarithms of the trigonometric functions, one gets a very nice periodic pattern (It's not necessary to use absolute values, the pattern is the same if we only use $N$ which return positive values).
What causes these patterns? Why is $\cos N$ 'half a period' behind $\sin N$ in its pattern? Are the numbers $N$ closest to the 'intersections' in any way special?
Or is this just Mathematica computation failure?
The patterns you are seeing are because some integers are really close to multiples of $\pi$. For example, about every 22 elements of your set are really close to 0. This is because $22\approx 7\pi$. Thus, $$\log(|\sin(11)|)\approx\log(|\sin(7\pi/2)|)=0,$$ and $$\log(|\sin(33)|)\approx\log(|\sin(21\pi/2)|)=0,$$ and so forth. This accounts for the smallest "periodic" pattern I see. (I put periodic in quotes because it is not really periodic, only approximately so). The other, wider patterns correspond to more accurate rational approximations of $\pi$, like 355/113.