Polar Form of Complex Conjugate

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Theorem

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


Then:

$\overline z = r \paren {\cos \theta - i \sin \theta}$

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


Proof

\(\ds z\) \(=\) \(\ds r \paren {\cos \theta + i \sin \theta}\)
\(\ds \) \(=\) \(\ds \paren {r \cos \theta} + i \paren {r \sin \theta}\)
\(\ds \leadsto \ \ \) \(\ds \overline z\) \(=\) \(\ds \paren {r \cos \theta} - i \paren {r \sin \theta}\) Definition of Complex Conjugate
\(\ds \) \(=\) \(\ds r \paren {\cos \theta - i \sin \theta}\)

$\blacksquare$


Sources