Circuit Theory/Phasors/proof7

$g(t)=G_{m}cos(\omega t-\phi )$

$g(t)=G_{m}\operatorname {Re} (e^{j(\omega t-\phi )})$

$g(t)=G_{m}\operatorname {Re} (e^{-j*\phi }e^{j\omega t})$

$g(t)=\operatorname {Re} (G_{m}e^{-j*\phi }e^{j\omega t})$

$g(t)=\operatorname {Re} (\mathbb {G} e^{j\omega t})$

$\mathbb {G} =G_{m}e^{-j*\phi }=G_{m}(cos(-\phi )+j*sin(-\phi ))=G_{m}cos(\phi )-jG_{m}sin(\phi )$