Answer
The required ordered pair in polar coordinates is $\left( 4,\frac{5\pi }{3} \right)$.
Work Step by Step
The Cartesian coordinates in polar form are $x=r\,\cos \,\theta \ \ \text{ and }\ y=r\,\sin \,\theta $.
The provided data is $x=2\ \text{ and }\ y=-2\sqrt{3}$.
Therefore, the required polar coordinates can be determined by using the following formulas:
$r=\sqrt{{{y}^{2}}+{{x}^{2}}}\ \text{ and }\ \text{tan}\,\theta =\frac{y}{x}$
Therefore, the polar coordinate r is given by:
$\begin{align}
& r=\sqrt{{{y}^{2}}+{{x}^{2}}} \\
& =\ \sqrt{{{2}^{2}}+{{\left( -2\sqrt{3} \right)}^{2}}} \\
& =4
\end{align}$
and
$\begin{align}
& \text{tan}\,\theta =\frac{y}{x} \\
& =\frac{-2\sqrt{3}}{2} \\
& =-\sqrt{3}
\end{align}$
Here, the tan function is negative; thus, $\theta $ would lie in the fourth quadrant.
Therefore,
$\begin{align}
& \theta =2\pi -\frac{\pi }{3} \\
& =\frac{5\pi }{3}
\end{align}$
Thus, the required polar coordinates are $\left( r,\theta \right)=\left( 4,\frac{5\pi }{3} \right)$.
