[ChrisLomont]

I just went through this - you cannot. Sine is transcendental. For no rational inputs (other than 0) is the output rational. Sampling at discrete timesteps and quantizing is giving you not samples of a sine wave, but samples close to a sine wave. The reconstruction is not the original sine wave - it's no longer meeting Nyquist theorem requirements.

The oscilloscope can detect that the pattern of inputs has a frequency, but it's necessarily an approximation at this point. An oscilloscope adds enough noise and error from it's own workings that on a screen, to your eye, for certain sampling parameters, it looks close. But that is not the original signal.

Take a signal, sample it, DAC it, and try using that signal to cancel the original, amplify the result, and run that through an oscilloscope. If the signal were reconstructed, you should be able to get zero.

You don't. Now you see the differences in the reconstructed signal and the original.