Circuit Theory/Phasors/proof8

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

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

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

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

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

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

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

=G_{m}cos(-\phi -{\frac {\pi }{2}})+jG_{m}sin(-\phi -{\frac {\pi }{2}})

=G_{m}(cos(-\phi )cos({\frac {\pi }{2}})+sin(-\phi )sin({\frac {\pi }{2}}))+jG_{m}(sin(-\phi )cos({\frac {\pi }{2}})-cos(-\phi )sin({\frac {\pi }{2}}))

=G_{m}sin(-\phi )-jG_{m}cos(-\phi )

=-G_{m}sin(\phi )-jG_{m}cos(\phi )

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