Using Unit Circle to Find Sine, Cosine, and Tangent

For every positive integer $n$, define $S_n$ to be the sum $$S_n = \sum_{k = 1}^{2010} \left( \cos \frac{k! \pi}{2010} \right)^n .$$ As $n$ approaches infinity, what value does $S_n$ approach?

(Hint: For which values of $k$ is $\cos\frac{k! \pi}{2010} = \pm 1$?)

Source: Math Prize for Girls