http://en.wikipedia.org/wiki/Imaginary_unit#Powers_of_i
These are a couple basic "identities"--by which I mean they're playing around with 'i to a power' to illustrate patterns. You can verify any of them for yourself pretty easily with some algebra and possibly a little scratching of the head. (Ask if you're having trouble.) Remember that:
BLAH ^ (-1) = 1/BLAH
If you're looking for what USES imaginary numbers have, most of electrical engineering would fall hard on its face without the concept of i (except EEs call it j instead of i for some reason I don't care to investigate). Scroll down a little on the link I gave to see Euler's beautiful formulas for a cursory glance at WHY that's the case. (Don't expect to fully appreciate or even be able to apply the equations unless you have a little background in stuff like that.)