Polar Form of Complex Conjugate

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Theorem

Let $z := r \left({\cos \theta + i \sin \theta}\right) \in \C$ be a complex number expressed in polar form.


Then:

$\overline z = r \left({\cos \theta - i \sin \theta}\right)$

where $\overline z$ denotes the complex conjugate of $z$.


Proof

\(\displaystyle z\) \(=\) \(\displaystyle r \left({\cos \theta + i \sin \theta}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({r \cos \theta}\right) + i \left({r \sin \theta}\right)\)
\(\displaystyle \implies \ \ \) \(\displaystyle \overline z\) \(=\) \(\displaystyle \left({r \cos \theta}\right) - i \left({r \sin \theta}\right)\) Definition of Complex Conjugate
\(\displaystyle \) \(=\) \(\displaystyle r \left({\cos \theta - i \sin \theta}\right)\)

$\blacksquare$


Sources