Trigonometric Equations?

Question

How many solutions does the equation sin x = 1/810 x have where x is a number of degrees?

Answer 1

The easiest way to solve this is by graphing. Since |sin(x)| ≤ 1, there can be no solutions for x > 810 or x < -810 (because | x/810 | > 1 for these ranges). So graph y=sin(x) for 0 ≤ x ≤ 810 (just a quick sketch will do -- plot sin(x) at x=0,90,180,...,810 and sketch in the curve; and obviously, don't use the same scale for the x and y axes!). The graph of y=x/810 is a line connecting the points (0,0) and (810,1).

Now look for intersections of the line and the sine. I count 5: one at x=0 and four for positive values of x. Since both sin(x) and x/810 are odd functions, there are also four negative solutions (the negatives of the four positive solutions), for a total of 9 solutions. Or if you prefer not to use the "odd function" argument, just draw your graphs for x from -810 to 810 (the line goes from
(-810,-1) through (0,0) to (810,1)).