4040.11 – When Does sin(x) = x/100?

Tags:Problem Set 2Pre-Calculus MathTrigonometry

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


This problem is great for a graphing calculator.

Because sinx1|\sin x| ≤ 1, we need to look for values of xx such that x1001|\frac{x}{100}| ≤ 1, that is, x100|x| ≤ 100. In each period of 2π2\pi, from 0 to 100, there are two intersections of f(x)=sinxf(x) = \sin x and g(x)=x/100g(x) = x/100, including the one at x=0x = 0.

We know 100/2π100/2\pi is about 15.915.9, 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.