Polar Coordinates

Intuition and core definition

Polar coordinates locate a plane point by distance and direction: $(r,\theta)$ means move signed distance $r$ along the ray at angle $\theta$. Unlike Cartesian coordinates, polar representations are not unique: adding full turns or using negative radius with angle shifted by $\pi$ gives the same point.

Notation, language, and conditions

Conversions are $x=r\cos\theta$, $y=r\sin\theta$, and $r^2=x^2+y^2$. When $x\ne0$, $\tan\theta=y/x$ gives a reference value but still needs quadrant correction; $\operatorname{atan2}(y,x)$ also handles $x=0$. Usually $r\ge0$ for a principal representation, but negative $r$ is allowed in general. At the origin, angle is arbitrary.

Why this idea matters

Polar coordinates locate a point by distance and direction, often simplifying geometry organized around a centre or rotation.

A dependable method

1. For polar-to-Cartesian, compute cosine and sine components with correct angle units.
2. For Cartesian-to-polar, find $r=\sqrt{x^2+y^2}$.
3. Determine quadrant from signs of $x,y$ and choose an appropriate argument.
4. State the requested range for $\theta$ and whether $r$ must be nonnegative.
5. Convert back to check both coordinates.

Worked example

Common mistakes