math confusion ?

It may be the late hour, but the following, seemingly trivial confusion caused me grief for 15 minutes. I know it's serious, since I asked a collegue (who's VERY smart), and he also got confused. Here it is:

• f is frequency - cycles/sec
• one cycle equals 2pi radians
• therefore, f = 2pi*rad / sec
• rad / sec is angular velocity - w
• therefore, f = 2pi*w
• But it's a well known fact/formula that w = 2pi*f

WTF ?

I'll post the answer a bit later.