# Polar Form of Reciprocal of Complex Number

## Theorem

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

Then:

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

## Proof

 $\displaystyle \dfrac 1 z$ $=$ $\displaystyle \dfrac {\overline z} {z \overline z}$ Inverse for Complex Multiplication $\displaystyle$ $=$ $\displaystyle \dfrac {r \left({\cos \theta - i \sin \theta}\right)} {r \left({\cos \theta + i \sin \theta}\right) r \left({\cos \theta - i \sin \theta}\right)}$ Polar Form of Complex Conjugate $\displaystyle$ $=$ $\displaystyle \dfrac {\left({\cos \theta - i \sin \theta}\right)} {r \left({\cos^2 \theta - i^2 \sin^2 \theta}\right)}$ simplifying $\displaystyle$ $=$ $\displaystyle \dfrac {\left({\cos \theta - i \sin \theta}\right)} {r \left({\cos^2 \theta + \sin^2 \theta}\right)}$ Definition of Imaginary Unit $\displaystyle$ $=$ $\displaystyle \dfrac {\cos \theta - i \sin \theta} r$ Sum of Squares of Sine and Cosine

$\blacksquare$