Let's suppose that we have a complex number with
r=1⟹z=cosϕ+isinϕ
Then why is z−1=cos(−ϕ)+isin(−ϕ)=cosϕ−isinϕ
Answer
This is De Moivre's formula
in action:
For complex z and any integer n we have
zn=(cos(φ)+isin(φ))n=cos(nφ)+isin(nφ)
The formula
z−1=cos(−φ)+isin(−φ)
is De Moivre's formula evaluated at n=−1.
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