If you want to work it out yourself first here is the trick. Set calculator to degree mode, enter a bunch of fives (5555 or 55555555), take the reciprocal, take the sin, note similarity to pi. Very fun article.
pi / 180 = 0.01745 (a value commonly used for degree<->radian conversion). 1 / 180 = 0.0055555. That's the connection between the "magic" constants used in this trick.
It brings an older, simpler, not nearly as interesting calculator "trick" to mind: 11111111 * 11111111.