Answer
.
Work Step by Step
The provided complex number $z=a+ib$ is in rectangular form with $a$ and $b$ as its real and imaginary parts respectively.
First find the modulus of the complex number, which is represented by $r$ as shown below:
$r=\sqrt{{{a}^{2}}+{{b}^{2}}}$
Then find the argument of the complex number, which is represented by $\theta $ as shown below:
$\begin{align}
& \tan \theta =\frac{b}{a} \\
& \theta ={{\tan }^{-1}}\frac{b}{a}
\end{align}$
Substitute the value of the modulus $\left( r \right)$ and argument $\left( \theta \right)$ in the equation provided below:
$\begin{align}
& z=r\left( \cos \theta +i\sin \theta \right) \\
& =\sqrt{{{a}^{2}}+{{b}^{2}}}\left\{ \cos \left( {{\tan }^{-1}}\frac{b}{a} \right)+i\sin \left( {{\tan }^{-1}}\frac{b}{a} \right) \right\}
\end{align}$
Above is the polar form of the assumed complex number.
Example:
Above explanation can be justified with the help of an example. Let the provided complex number be $z=3+i4$ in simple rectangular form. To convert this number into polar form, first find the modulus and argument of the same number.
$\begin{align}
& r=\sqrt{{{a}^{2}}+{{b}^{2}}} \\
& =\sqrt{{{3}^{2}}+{{4}^{2}}} \\
& =\sqrt{25} \\
& =5
\end{align}$
Modulus of the provided number is $5$.
The argument of the complex number is,
$\begin{align}
& \tan \theta =\frac{b}{a} \\
& =\frac{4}{3} \\
& \theta ={{\tan }^{-1}}\left( \frac{4}{3} \right) \\
& \approx 53.13{}^\circ
\end{align}$
The argument of the provided number is $53.13{}^\circ $
Represent provided complex number in polar form,
$z=5\left( \cos 53.13{}^\circ +i\sin 53.13{}^\circ \right)$
The above expression is the polar form of the provided complex number.
