An arbitrary [circle](https://artofproblemsolving.com/wiki/index.php/Circle) can intersect the [graph](https://artofproblemsolving.com/wiki/index.php/Graph) of $y=\sin x$ in
$\mathrm{(A) \ } \text{at most }2\text{ points} \qquad \mathrm{(B) \ }\text{at most }4\text{ points} \qquad \mathrm{(C) \  } \text{at most }6\text{ points} \qquad \mathrm{(D) \ } \text{at most }8\text{ points}\qquad \mathrm{(E) \  }\text{more than }16\text{ points}$
Consider a circle with center on the positive y-axis and that passes through the origin. As the radius of this circle becomes arbitrarily large, its shape near the x-axis becomes very similar to that of $y=0$, which intersects $y=\sin x$ infinitely many times. It therefore becomes obvious that we can pick a radius large enough so that the circle intersects the sinusoid as many times as we wish, $\boxed{\text{E}}$.