Have you thought about this at all?

Clearly we only need to consider . For each period of sin(x), the line y=x/100 will cross the curve y=sin(x) twice (starting at the origin), except that next to the origin it will only cross once, and be careful at the endpoints. Graph it if you are unsure.

In case you didn't follow what I just wrote: On the interval (0, 2pi] there is one intersection, on the interval [2pi, 4pi] there are two intersections, etc. On the interval [-2pi, 0) there is one intersection, on the interval [-4pi, -2pi] there are two intersections, etc.

And don't forget the intersection at the origin.

Also note that due to symmetry the number of intersections in (0,100] will be the same as the number of intersections in [-100,0).