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It's just as I said originally -- what you are seeing is a special case of "it works this time by coincidence".
You are mixing time-domain and frequency-domain components.
If your time-domain input signal is Vo·sin(wt), then this is
\(
V_0 sin \left( \omega t \right) \; = \; Im \left\{ V_0 e^{j \omega t} \right\}
\)
Now work with the complex signal and take the Imaginary component at the end and consider what happens if you multiply this by H = A + jB
\(
V_0 e^{j \omega t} \cdot H \\
V_0 e^{j \omega t} \cdot \left( A \; + \; jB \right) \\
V_0 \left( cos \left( \omega t \right) \; + \; j sin \left( \omega t \right) \right) \cdot \left( A \; + \; jB \right) \\
V_0 \left[ \left( A cos \left( \omega t \right) \; - \; B sin \left( \omega t \right) \right) \; + \; j \left( A sin \left( \omega t \right) \; + \; B cos \left( \omega t \right) \right) \right] \\
V_0 \left( A cos \left( \omega t \right) \; - \; B sin \left( \omega t \right) \right) \; + \; j V_0 \left( A sin \left( \omega t \right) \; + \; B cos \left( \omega t \right) \right)
\)
What are we left with when we take the imaginary part at the end to go back to the time domain?
\(
V_0 \left( A sin \left( \omega t \right) \; + \; B cos \left( \omega t \right) \right)
\)
So you can see that what you think you are observing is merely an artifact of the math in this problem.
I can't tell if your final step is correct because I don't know if you are going to use sin() or cos() for your final result.
