Explanation:
The polar form of a complex number z = x + iy is calculated using the formula below:
[tex]z=rcis\text{ }\theta[/tex]
where:
r = sqrt root of (x² + y²) and is also considered as the modulus of the complex number
θ = tan^-1 (y/x) is the argument of the complex number.
So, for the first complex number 3 - √3i, let's convert this to polar form in which x = 3 and y = - √3.
To convert, let's solve for the "r" or the modulus first.
[tex]\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{3^2+(-\sqrt{3})^2} \\ r=\sqrt{9+3} \\ r=\sqrt{12} \\ r=2\sqrt{3} \end{gathered}[/tex]
The modulus is 2√3.
Let's solve for the θ or the argument.
[tex]\begin{gathered} \theta=tan^{-1}(\frac{y}{x}) \\ \theta=tan^{-1}(\frac{-\sqrt{3}}{3}) \\ \theta=-\frac{1}{6}\pi \end{gathered}[/tex]
The argument is -1/6π.
Plugging in the modulus and argument to the abbreviated polar form of a complex number, the result is:
[tex]rcis\theta\Rightarrow2\sqrt{3}cis-\frac{1}{6}\pi[/tex]
