$3.$ Phasors and complex exponential function $(10$ points$)$

a$)$ Consider $2$ voltage functions:

v$1($t$)=5$ sin$(10$t $+60$o$)($V$)$ and v$2($t$)=5$ cos$(10$t $+30$o$)($V$)$

i$)$ Find angular frequency

and phasors V$1$ and V$2.$

ii$)$ For a RL circuit, we have phasor equation I $=$ V$1$ V$2$

Z$1$

where Z$1=$ R $+$ j

L$,$ R $=10()$ and L $=3($H$).$

Solve for phasor current I and find time varying current in terms of cos$.$

b$)$ i$)$ Express $3+$ i in exponential form with all possible values for angle,

ii$)$ For Z $=$ x $+$ iy$,$ find the magnitude and angle of e$5$z$1.$

iii$)$ Find all possible values of e$5$z$1=3+$ i$.$