Circuit Theory/Phasors/proof2

g(t)=G_{m}sin(\omega t)

starting point

g(t)=G_{m}cos(\omega t-{\frac {\pi }{2}})

g(t)=G_{m}\operatorname {Re} (e^{j(\omega t-{\frac {\pi }{2}})})

g(t)=G_{m}\operatorname {Re} (e^{-j*{\frac {\pi }{2}}}e^{j\omega t})

g(t)=\operatorname {Re} (G_{m}e^{-j*{\frac {\pi }{2}}}e^{j\omega t})

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

\mathbb {G} =G_{m}e^{-j*{\frac {\pi }{2}}}=G_{m}(cos(-{\frac {\pi }{2}})+j*sin(-{\frac {\pi }{2}}))=-jGm

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