For a sinusoid frequency of $1200$Hz and a sampling frequency of $2000$Hz and a reconstruction frequency of $2000$Hz and $3000$Hz, what frequency will the sinusoid be after the reconstruction?
I divided $1200/2000$ for $f/f_s$ but am unsure of how to apply the reconstruction frequency to get the answer. ... $1200/2000 \cdot 2\pi = 1.6\pi$ and for the reconstruction frequency $2\pi/3000$, what would I do to get the end result?
If you keep the units around, it makes it much easier. If you sample at $f_s$, then every $1/f_s$ seconds a sample is recorded. If the sinusoid you are sampling has a frequency of $2\pi f_0$ (rad/s) and you sample it as $f_s$, then we get
$$ 2\pi f_0 \mathrm{(rad/s)} \cdot { {1} \over {f_s} } \mathrm{(s/sample)} = 2\pi { {f_0} \over {f_s} } \mathrm{(rad/sample)} $$
If you reconstruct with a frequency of $f_r$, then every sample will be treated as if it is $1/f_r$ seconds apart, which means there will be $f_r$ samples per second. Thus, the reconstructed frequency will be
$$ 2\pi { {f_0} \over {f_s} } \mathrm{(rad/sample)} \cdot f_r \mathrm{ (samples/s) } = 2\pi f_0 { {f_r} \over {f_s} } \mathrm{(rad/sample)}. $$
The point of this exercise seems to be that your sampling frequency falls 400Hz short of the Nyquist rate of 2400Hz. Thus your sampled signal will be indistinguishable from a sine wave at 800Hz=(2000Hz-400Hz)/2. You should be able to work out the corresponding amplitudes and pahse shifts, if any.
