Parametric & polar

Polar coordinates describe a point by its distance $r$ from the origin and the angle $\theta$ measured from the positive $x$‑axis. To move between polar and rectangular (Cartesian) forms, use the relationships $x=r\cos \theta$, $y=r\sin \theta$, $r=\sqrt{x^2+y^2}$, and $\tan \theta =\frac{y}{x}$ (when $x\ne 0$). When converting a rectangular equation to polar, replace every $x$ with $r\cos \theta$ and every $y$ with $r\sin \theta$, then simplify if possible. Conversely, to turn a polar equation into Cartesian, replace $r$ with $\sqrt{x^2+y^2}$ and $\theta$ with the appropriate trigonometric expressions using $\cos \theta =\frac{x}{r}$ and $\sin \theta =\frac{y}{r}$. These substitutions let you rewrite the same curve in either coordinate system.