From my knowledge your professor might have just been wrong. Or maybe he used his own notation for the angular acceleration as omega, thing that I never saw in any textbook, which practically makes it wrong. There is nothing about the cancellation of the radian unit (it can be canceled for dimensional analysis purposes as of the "recent" physics congress) either, because we are talking about the powers of the second, so has to be one of the above from my perspective.

From my knowledge your professor might have just been wrong. Or maybe he used his own notation for the angular acceleration as omega, thing that I never saw in any textbook, which practically makes it wrong. There is nothing about the cancellation of the radian unit (it can be canceled for dimensional analysis purposes as of the "recent" physics congress) either, because we are talking about the powers of the second, so he just is probably wrong.