Answer to Question #86584 in Complex Analysis for nur

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=\\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=-\\frac{1}{2}j"

where


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

In polar form the complex number z is written as


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

where

"r=\\left| z \\right|=\\sqrt{{{x}^{2}}+{{y}^{2}}}=\\sqrt{{{0}^{2}}+{{\\left( -\\frac{1}{2} \\right)}^{2}}}=\\sqrt{\\frac{1}{4}}=\\frac{1}{2}""\\theta =\\arctan \\left( \\frac{y}{x} \\right)=\\arctan \\left( \\frac{-1\/2}{0} \\right)=\\arctan \\left( -\\infty \\right)=-\\frac{\\pi }{2}"


or otherwise

"\\theta =-\\frac{\\pi }{2}+2\\pi =\\frac{3\\pi }{2}"

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

"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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