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Write an equation that relates $\arg(z)$ to $\arg(1/z)$, $z\not = 0$.

I don't have a clear idea of how attack this type of exercise.

I know if $z=r(\cos\theta +i\sin\theta$) or equivalent to $z=x+iy$ then the $\arg(z)=\theta$ and we can calculate $\theta$ using the fact $\tan\theta=\frac{y}{x}\implies\theta=\tan^{-1}(\frac{y}{x})$ But, here I'm stuck. Can someone help me?

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2 Answers 2

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Write $z$ in the polar form as $z=re^{i\theta}$. Then $1/z=\frac{1}{r}e^{-i\theta}$. So $\arg(1/z)=-\arg(z)$.

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  • $\begingroup$ Even for $z=-1$? $\endgroup$ Oct 12, 2018 at 15:18
  • $\begingroup$ Yes, because $\arg(1/-1)=\arg(-1)=\pi=-\pi=-\arg(-1)$. The angles $\pi$ and $-\pi$ are equivalent. Although you do bring up a good point, which is that you may need to add or subtract $2\pi$ to get the argument in the usual range $\endgroup$
    – HackerBoss
    Oct 12, 2018 at 15:49
  • $\begingroup$ Ah $\pi=-\pi$. You live and learn! $\endgroup$ Oct 12, 2018 at 15:50
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So if $$z = r(\cos \theta + i \sin \theta)$$ what is the expression for $1/z$? You can use the usual trick for division by complex numbers: multiple numerator and denominator by the complex conjugate.

Now once you have an expression for $1/z$, you should be able to infer the correct angle.

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