## Precalculus (6th Edition) Blitzer

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.