Answer
$r=4$.
Work Step by Step
Rectangular equation is ${{x}^{2}}+{{y}^{2}}=16$ …… (I)
The relation between polar coordinates and rectangular coordinates can be expressed as below:
$x=r\cos \theta \ \text{ and }\ y=r\sin \theta $ …… (II)
Substitute values of $x\ \text{ and }\ y$ from (II) in (I), to get
$\begin{align}
& {{x}^{2}}+{{y}^{2}}=16 \\
& {{\left( r\cos \theta \right)}^{2}}+{{\left( r\sin \theta \right)}^{2}}=16 \\
& {{r}^{2}}\left( {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \right)=16
\end{align}$
As,
$\left( {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \right)=1$
From here we get,
$\begin{align}
& {{r}^{2}}=16 \\
& r=4
\end{align}$
Hence the obtained polar expression is $r=4$.
