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

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?

Write $$z$$ in the polar form as $$z=re^{i\theta}$$. Then $$1/z=\frac{1}{r}e^{-i\theta}$$. So $$\arg(1/z)=-\arg(z)$$.
• Even for $z=-1$? – Lord Shark the Unknown Oct 12 '18 at 15:18
• 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 – HackerBoss Oct 12 '18 at 15:49
• Ah $\pi=-\pi$. You live and learn! – Lord Shark the Unknown Oct 12 '18 at 15:50
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.