Polar coordinates that uses $\frac { 1 }{ Z_1 }$

I am doing polar coordinates, and I am stuck when my book asks to do $\frac { 1 }{ Z_1 }$. I have no problems with $\frac { Z_1 }{ Z_2 }$ and $Z_1Z_2$. Here is the values for $Z_1$ I'm not so much concerned with the answer more how to do it.

$$Z_1 = \sqrt{3} + i$$ $$r = 2, \theta = \frac {\pi}{6}$$ $$\frac { 1 }{ Z_1 }$$

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2 Answers

$\frac{1}{z} = \frac{\overline{z}}{\overline{z}} \frac{1}{z} = \frac{\overline{z}}{|z|^2}$.

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$$z_1=2\,e^{\frac{\pi i}{6}}\Longrightarrow \frac{1}{z_1}=z_1^{-1}=2^{-1}e^{-\frac{\pi i}{6}}=\ldots$$

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