## Precalculus (6th Edition) Blitzer

The polar form of a complex number is yet another form to represent a complex number, other than rectangular form. A complex number $z=a+ib$ can be represented in polar form as: $z=r\left( \cos \theta +i\sin \theta \right)$ Here $a=r\cos \theta$, $b=r\sin \theta$, $r=\sqrt{{{a}^{2}}+{{b}^{2}}}$ and $\tan \theta =\frac{b}{a}$ The values $r$ and $\theta$ are called the modulus and argument of a complex number $z$ respectively, with $0\le \theta \le 2\pi$. Example: The above explanation can be justified with the help of an example. The provided complex number $z=5\left( \cos 30{}^\circ +i\sin 30{}^\circ \right)$ is in simple polar form.The modulus and argument of a complex number is $5$ and $30{}^\circ$ respectively. Simplify the complex number, \begin{align} & z=5\left( \cos 30{}^\circ +i\sin 30{}^\circ \right) \\ & =5\left( \frac{\sqrt{3}}{2}+i\frac{1}{2} \right) \\ & =\frac{5\sqrt{3}}{2}+i\frac{1}{2} \end{align} The real part of a complex number is $\frac{5\sqrt{3}}{2}$ and the imaginary part of a complex number is $\frac{1}{2}$. The above expression of a complex number in polar form is converted into rectangular form.