Circuit Theory/Phasors/proof5

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