I have a time series (of 23 timestamps) of which I take the Fourier transform. Now the fourier transform has 23 imaginary values and each has an amplitude and a phase. When I get the phase angle, it given me arctan value of the angle. Now how can I determine if two fourier coefficients are in phase or out of phase.
Depends on the phase reference point in time.
Depending on the reference point, every DFT basis sinusoid can be either in-phase or out-of-phase (by 180 degrees) with every other basis, thus the same with any coefficient weighting of these basis vectors.
If you reference phase to the center of a DFT (using an fftshift), then the angle will tell you whether a particular component has even or odd symmetry relative to the center.