Converting between polar and rectangular

$x=r\cos\theta,\ y=r\sin\theta$ one way; $r^2=x^2+y^2,\ \tan\theta=y/x$ the other.

Converting FROM polar TO rectangular uses $x=r\cos\theta$ , $y=r\sin\theta$ directly. Converting FROM rectangular TO polar uses $r^2=x^2+y^2$ (so $r=\sqrt{x^2+y^2}$ ) and $\tan\theta=y/x$ — but $\tan\theta=y/x$ alone doesn't determine the QUADRANT, so the point's actual $(x,y)$ signs must also be checked to pick the correct $\theta$ .

Examples

Rectangular $(-1,1)$ : $r=\sqrt{2}$ , and $\tan\theta=-1$ suggests $\theta=-\pi/4$ or $3\pi/4$ — checking the signs, $(-1,1)$ is in Quadrant II, so $\theta=3\pi/4$ .