Circuit Theory/Phasors/proof1

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g(t)=G_m cos(\omega t) starting point
g(t)=G_m \operatorname{Re}(e^{j\omega t})
g(t)=G_m \operatorname{Re}(e^{j*0}e^{j\omega t})
g(t)=\operatorname{Re}(G_m e^{j0}e^{j\omega t})
g(t)=\operatorname{Re}(\mathbb{G} e^{j\omega t})
\mathbb{G} = G_m e^{j0} = G_m(cos(0) + j*sin(0)) = Gm