Polar Form of Reciprocal of Complex Number

Theorem

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

Then:

$\dfrac 1 z = \dfrac {\cos \theta - i \sin \theta} r$

$\ds \dfrac 1 z$

$=$

$\ds \dfrac {\overline z} {z \overline z}$

$\ds $

$=$

$\ds \dfrac {r \paren {\cos \theta - i \sin \theta} } {r \paren {\cos \theta + i \sin \theta} r \paren {\cos \theta - i \sin \theta} }$

$\ds $

$=$

$\ds \dfrac {\paren {\cos \theta - i \sin \theta} } {r \paren {\cos^2 \theta - i^2 \sin^2 \theta} }$

simplifying

$\ds $

$=$

$\ds \dfrac {\paren {\cos \theta - i \sin \theta} } {r \paren {\cos^2 \theta + \sin^2 \theta} }$

Definition of Imaginary Unit

$\ds $

$=$

$\ds \dfrac {\cos \theta - i \sin \theta} r$

$\blacksquare$

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: $(2.14)$