Polar Form of Reciprocal of Complex Number

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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

$\blacksquare$


Sources