Polar Form of Complex Conjugate

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

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations