The solution is very simple, actually. There's a units confusion in **f**:

The units of f is Hz, namely 1/sec. f is a physical quantity, radians and cycles are units, we can't put quantity and units in the same equation !

So, if we say that in each second 1 cycle goes, it means that 2pi radians go, so:

w = 2pi*f (we can put w and f in the same equation).

Example: think that f = X Hz:

f = X [units: 1/sec]

X cycles a second => 2pi*X radians a second => w = 2pi*X [units: rad/sec]