My intuition is that the $2\pi$ term in the sinusoidal signal equation:
$$x(t) = \sin(2\pi\,f\,t)$$
Is indicative of the fact that this signal can be described as movement around a circle, is that correct?
And the sin component is describing the path it takes about that circle?
*Due to a lack of reputation I'm unable to properly tag this inquiry, would someone please help me remedy that.
The $2\pi$ in the equation is to make the $f$ represent cycles per second (assuming $t$ is in seconds). For example, if $f=10$, then the sine wave $\sin(2\pi\,10\,t)$ will crest 10 times from $t=0$ to $t=1$.
Circles can always be used to represent sine waves. The height of a person on a Ferris wheel is is a sine wave, even though their motion is a circle. See the top answer to this question.