Do you need an infinite number of points to extrapolate a sinusoid?

So I was thinking this through today, just because I was curious. Following the logic of you need 2 points to find a line, 3 points to find a quadriatic, 4 points for a cubic, etc, I was trying to figure out if its possible to find a unique sinusoid using a finite number of points. I figured that, if you had a particular sinusoid that went through a set of points, you can likely increase the frequency and add a slight phase shift to get another solution that goes through the same set of points. But then, the part that I'm not certain on, is I figured if you placed the points you used at an interval that is an irrational multiple of the frequency of your original solution, you wouldn't be able to find higher frequency solutions. Is there any validity to this? Is there a definite number of points you need to be able to extrapolate a unique solution?