Question #86584
Question: please convert this complex number into the polar form.
1/2j

note: it is not (1/2) * j
it is 1/2*j --> j is at denominator
1
Expert's answer
2019-03-20T14:46:50-0400

Multiply the numerator and denominator of this complex number by j


z=12j=12jjj=j2j2=j2(1)=12jz=\frac{1}{2j}=\frac{1}{2j}\cdot \frac{j}{j}=\frac{j}{2{{j}^{2}}}=\frac{j}{2\cdot \left( -1 \right)}=-\frac{1}{2}j

So in a rectangular form we have


z=x+jy=12jz=x+jy=-\frac{1}{2}j

where


x=0,y=12x=0,\,\,y=-\frac{1}{2}

In polar form the complex number z is written as


z=r(cosθ+jsinθ)z=r\left( \cos \theta +j\sin \theta \right)

where

r=z=x2+y2=02+(12)2=14=12r=\left| z \right|=\sqrt{{{x}^{2}}+{{y}^{2}}}=\sqrt{{{0}^{2}}+{{\left( -\frac{1}{2} \right)}^{2}}}=\sqrt{\frac{1}{4}}=\frac{1}{2}θ=arctan(yx)=arctan(1/20)=arctan()=π2\theta =\arctan \left( \frac{y}{x} \right)=\arctan \left( \frac{-1/2}{0} \right)=\arctan \left( -\infty \right)=-\frac{\pi }{2}


or otherwise

θ=π2+2π=3π2\theta =-\frac{\pi }{2}+2\pi =\frac{3\pi }{2}

Substituting the obtained values, we get the polar form of this z

z=12(cos(3π2)+jsin(3π2))z=\frac{1}{2}\left( \cos \left( \frac{3\pi }{2} \right)+j\sin \left( \frac{3\pi }{2} \right) \right)


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