## Precalculus (6th Edition) Blitzer

To convert a point from rectangular to polar coordinates or from $\left( x,y \right)$ to $\left( r,\theta \right)$ we will use the formula $r=\sqrt{{{x}^{2}}+{{y}^{2}}}$, for determining the value of r, and $\tan \theta =\frac{y}{x}$, for determining the value of $\theta$.
For example, consider a point $\left( 1,\sqrt{3} \right)$. It can be converted into polar coordinates as, \begin{align} & r=\sqrt{{{1}^{2}}+{{\left( \sqrt{3} \right)}^{2}}} \\ & =\sqrt{1+3} \\ & =\sqrt{4} \\ & =2 \end{align} \begin{align} & \tan \theta =\frac{\sqrt{3}}{1} \\ & \theta ={{\tan }^{-1}}\left( \sqrt{3} \right) \\ & =\frac{\pi }{3} \end{align} Hence, the rectangular point $\left( 1,\sqrt{3} \right)$ becomes $\left( 2,\frac{\pi }{3} \right)$ in polar coordinates.