1985 AHSME Problem 10

An arbitrary circle can intersect the 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}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.