Polar Coordinates

Definition

Polar coordinates are a coordinate system for \(\mathbb{R}^2\) in which a point is defined by its distance from the origin and the angle to a reference direction. In particular, we represent coordinates as the image under the map

\[ F(r, \theta) = (r \cos(\theta), r \sin(\theta)).\]

Theorem

Given coordinates of the form \((x, y) = (r \cos(\theta), r \sin(\theta))\), \(r\) is uniquely defined up to sign, and if \(r \neq 0\), then \(\theta\) is uniquely defined up to an integer multiple of \(2\pi\).

Proof

Suppose that \(x = r \cos(\theta)\) and \(y = r\sin(\theta)\). It is clear that

\[ \left(\frac{x}{r}\right)^2 + \left(\frac{y}{r}\right)^2 = \frac{x^2 + y^2}{r^2} = 1 \implies r^2 = x^2 + y^2 \implies r \in \{\pm \sqrt{x^2 + y^2}\}.\]

Now, suppose that there are two angles \(\theta\) and \(\varphi\) such that \(x = r\cos(\theta) = r\cos(\varphi)\) and \(y = r\sin(\theta) = r\sin(\varphi)\). Then, we have that

\[\begin{align*} \cos(\theta - \varphi) &= \cos(\theta)\cos(\varphi) + \sin(\theta)\sin(\varphi) \\ &= \frac{x^2}{r^2} + \frac{y^2}{r^2} \\ &= \frac{x^2 + y^2}{r^2} \\ &= 1. \\ \end{align*}\]

Hence \(\theta - \varphi = 2\pi n\).

From the above, it is typical to restrict to \(r \geq 0\) and \(\theta \in [0, 2 \pi)\), and enforce if \(r = 0\) then \(\theta = 0\), as to construct uniquely defined coordinates.