How many periods of a periodic signal are required to properly determine magnitude?
I have a pure sinusoidal test function.
  
  I already have a sufficiently high enough SNR to not need to worry too
  much about noise


If you know the frequency (as you say you do), then any fractional part of the sine wave can be analysed to give you the peak and RMS values of that pure sine wave. It's all in the math. 

No fractional part of a sine wave with amplitude X can be mistaken for a fractional part of another sine wave (amplitude Y) unless Y = X.

This means if noise is really low you can predict the amplitude from a few microseconds worth of signal.

Think what happens when the sine wave passes through zero - if you measured the slope it can tell you the amplitude because: -

\$\dfrac{d(A\cdot Sin (\omega t))}{dt} = \omega A\cdot Cos(\omega t)\$