Frequency of signal

The general form for a sinusoid $x(t)$ is

$$sin(2{pi}ft + phi)$$

Where $f$ is the frequency of the sinusoid and $phi$ is some constant phase, which many times is set to zero. This general expression makes the $2pi$-periodicity of the signal clear.

Questions that ask you to find the frequency of a simple sinusoid can be solved by setting $2{pi}ft$ equal to whatever is in the argument of the sinusoid you're dealing with. People get too comfortable with seeing the $pi$ term, so when it's no longer there it may throw you off.

The solution for the frequency $f$ is definitely more intuitive in the case of

$$2{pi}ft = 2{pi}t Rightarrow f = 1 text{ Hz}$$

Than

$$2{pi}ft = 2t Rightarrow f = frac{1}{pi} text{ Hz}$$

Both answers are just constant numbers and whether or not you see $pi$ is irrelevant, it is "built in" in to whatever expression is in the argument.

I have signal like that $ sin(2t) $ frequency of this signal is $ f = frac 1 pi $ but I am confused here because this $ pi $ equal 3.14 or 180° if signal is $ sin(2 pi t) $ then $ f = 1 $ and everything is clear.

Yes, whoever is setting the question is making the maths easy. One cycle per second or 1 Hz.