Let's say we have some 5 different sine waves, each having a frequency of 10 Hz but differing in phase i.e. 0, Pi/6, Pi/4, Pi/3, and Pi/2. Each of these signals is sampled at twice the signal frequency, therefore at 20 Hz.
Now upon the reconstruction of the samples, we notice that the amplitude of the reconstructed signal increases as the phase increases. When the phase is 0, the signal is essentially zero and when the amplitude is Pi/2, the signal amplitude is that of the original signal.
Now I can see why this happens by putting test values into the sine samples. For example, for the sine wave with 0 phase, all the samples at 20 hertz are essentially 0 and therefore the reconstruction is poor. I am trying to understand the general idea or a mathematical notion behind this. Is there something that I am missing from the sampling theorem?