Correct my frequency/period conversion (please)

Is frequency the inverse of period, or is it 2pi/period?

If you have an equation like this, A*sin(fx-p), or A*cos(fx-p), then f is the frequency (A is amplitude, p is phase shift). Allegedly, the period is T=2pi/f. Supposedly, frequency is f=1/T. Does anyone see anything wrong here (e.g, substituting 1/T for f in the equation for T does not yield T)?

For example y=sin(2t). Frequency f=2. Period T=2pi/f=2pi/2=pi (this matches my calculator which shows one cycle every pi). So, of course, f=1/T=1/Pi, except that f=2. What's the deal?