When sine of x degrees equals sine of x radians

Ordinarily the sine of x radians and the sine of x degrees are very different numbers. Having your calculator in radian mode when it should be in degree mode, or vice versa, results in a major error. But sometimes it doesn’t matter. A trivial example is when x = 0. A more interesting example is x = 180π/(180 + π) = 3.08770208…. For that vale of x, sin(x) = sin(x°). In this article I’ll use the common convention of using radians by default and denoting degrees with ° as above. Note that x° = πx/180 and so we are interested in solutions to the equation sin(x) = sin(πx/180) Now two angles A and B have the same sine if they differ by a multiple of 2π, or if they’re supplementary (i.e. A = π − B), or both. To put it another way, if A and B have the same sine, they are either equal mod 2π or supplementary mod 2π. This means that sin(x) = sin(πx/180) if and only if x = πx/180 + 2πk or x = π − πx/180 + 2πk for some integer k. Therefore all solutions have the form x = 360πk/(180 − π) or x =…