1981 AHSME Problems/Problem 18

Problem:

The number of real solutions to the equation is

Solution:

The answer to this problem is the number of intersections between the graph of f(x) = \sin x and f(x) = (1/100)x. We can do the right side of the coordinate plane first. Each cycle of the sine wave, consisting of 2π, will have 2 intersections (From the positive part of the sine wave) The line f(x) = (1/100)x will consist of 16 cycles plus a little bit extra for x from 1 to 100. However, the extra is not complete enough to have any intersection at all. Thus, the number of intersections is 2 x 16 = 32. Because of symmetry, we can multiply by two to account for the left side, and subtract one because of the origin. So the answer is 32 x 2 - 1 =

~Eric X