4040.11 – When Does sin(x) = x/100?
How many solutions are there of the equation ?
This problem is great for a graphing calculator.
Because , we need to look for values of such that , that is, . In each period of , from 0 to 100, there are two intersections of and , including the one at .
We know is about , so there are 16 pairs of intersections between 0 and 100. The 16th cycle, though not complete, does include the 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.