Circuit Theory/Phasors/proof2

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g(t)=G_m sin(\omega t) starting point
g(t)=G_m cos(\omega t - \frac{\pi}{2})
g(t)=G_m \operatorname{Re}(e^{j(\omega t - \frac{\pi}{2})})
g(t)=G_m \operatorname{Re}(e^{-j*\frac{\pi}{2}}e^{j\omega t})
g(t)=\operatorname{Re}(G_m e^{-j*\frac{\pi}{2}}e^{j\omega t})
g(t)=\operatorname{Re}(\mathbb{G} e^{j\omega t})
\mathbb{G} = G_m e^{-j*\frac{\pi}{2}} = G_m(cos(-\frac{\pi}{2}) + j*sin(-\frac{\pi}{2})) = -jGm