Circuit Theory/Phasors/proof5

$g(t)=G_{m}cos(\omega t+\varphi )$

$g(t)=G_m\mathrm{Re}(e^{j(\omega t+\varphi )})$

$g(t)=G_m\mathrm{Re}(e^{j*\varphi }{e}^{j\omega t})$

$g(t)=\mathrm{Re}(G_m{e}^{j*\varphi }{e}^{j\omega t})$

$g(t)=\mathrm{Re}(𝔾e^{j\omega t})$

$𝔾=G_m{e}^{j*\varphi }=G_m(cos(\varphi )+j*sin(\varphi ))=G_{m}cos(\varphi )+j{G}_{m}sin(\varphi )$