How do you convert #-3+1i# to polar form?

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Answer 1 · 2018-08-09T06:52:54

Please see below.

Let ,

#z=x+iy=-3+1i=>x=-3 and y=1#

Now ,

#|z|=r=sqrt(x^2+y^2)=sqrt((-3)^2+(1)^2)=sqrt(9+1)=sqrt10#

We have ,

#costheta=x/r=(-3)/sqrt10 < 0 and sintheta=y/r=1/sqrt10 > 0#

#:.costheta < 0 and sintheta > 0=>2^(nd)Quadrant#

#:.tantheta=sintheta/costheta=((1/sqrt10)/(-3/sqrt10))=-1/3#

#:.theta=arctan(-1/3)=-arc tan(1/3)~~(-18.43)^circ#

So, the polar form :

#z=r(costheta+isintheta)#

#=>z=sqrt10(costheta+isintheta)#

,where ,

#theta=-arc tan(1/3) ,costheta=-3/sqrt10 and sintheta=1/sqrt10 .#