Sin x = x/100

How many solutions are there for the equation sin x = x/100?


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CORRECTION: We thank Dan Showalter for catching an error in our solution to this problem. We believe the correct solution (now shown below) is 63 intersections, rather than the 59 intersections we originally indicated. Thanks, Dan!

Because |sin x| ≤ 1, we need to look for values of x such that |x/100| ≤ 1, that is, |x| ≤ 100. For each period of 2pi, from 0 to 100, there are two intersections (we include the one at x = 0) of f(x) = sin x and g(x) = x/100.

We know 100/2pi is about 15.9, so there are 16 pairs of intersections between 0 and 100 (the 16th cycle, though not complete, does include 2 intersections). Likewise, there are 16 pairs of intersections between -100 and 0, so that makes 32 pairs of intersections, or 64 solutions. Wait, we can’t count 0 twice, so there are 63 solutions.

(Great for a graphing calculator.)

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