Circuit Theory/Phasors/proof6

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