If I write in QM at an instant, that $\psi(x)=\cos(6\pi x)$. Does that mean $|\psi(x)|^2dx$ after normalisation gives me the probability that particle will be positioned between $x$ and $dx$ ?

If I write in QM at an instant, that the quantum state that describes the particle completely at an instant $\psi(x)=\cos(6\pi x)$. Does that mean $|\psi(x)|^2dx$ after normalisation gives me the probability that particle will be positioned between $x$ and $dx$ ? What if instead of position it was a function for state of momentum/energy, could I apply born's rule for getting wave function for position ?