I'm reading "Understanding Digital Signal Processing, 3rd Edition" by Richard Lyons. Chapter 9 derives Hilbert transform impulse response by defining it in frequency domain first and then taking inverse Fourier transform. I believe I could understand most of the derivation except one part.
It is stated that in order to get a -90° phase shift we have to multiply negative frequencies by $+j$ and positive frequencies by $-j$ in frequency domain:
The following Figure 9-3 clarifies it by considering the case when the source signal is a pure $\cos(\omega t)$:
It makes sense so far. Indeed $\cos(\omega t)$ consist of $\pm \omega$ frequencies with zero phase shift - exactly as it's shown in the top right corner. Multiplying by +j always gets us +90° rotation in the complex plane and multiplying by -j is always equal to rotating -90°, regardless to the sign of the frequency. Frequency in this case is a third axes, completely unrelated to the complex plane (Re-Im).
However, I don't understand the reason why the result (the right bottom plot) is a sine wave in time domain (the left bottom plot). This must me something trivial I guess, but unfortunately I'm new to DSP and not a mathematician. Could you please explain this step?