# Polar Form of Complex Conjugate

## 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 $\displaystyle$ $=$ $\displaystyle r \left({\cos \theta - i \sin \theta}\right)$

$\blacksquare$