3.1 Polar coordinates

Let $(x,y)$ be a point different from $(0,0)$; we can specify its polar co–ordinates as follows. We draw the circle with centre $(0,0)$ that passes through $(x,y)$, and let $r>0$ be its radius and $\theta\in[0,2\pi)$ be the angle at $(0,0)$ between the positive real axis and the direction to $(x,y)$. The formulæ

follow from the geometric definitions of the trigonometric functions. We can recover $r$ from Pythagoras’ theorem $r^{2}=x^{2}+y^{2}$ and then determine $\theta\in[0,2\pi)$ uniquely.

$\bullet$ Fix $r$. Then $(x,y)$ describes the circle of centre $(0,0)$ and radius $r$ as $\theta$ ranges over $[0,2\pi)$.

$\bullet$ Fix $\theta$. As $r$ increases from $0$, the point $(x,y)$ describes the straight line through $(0,0)$ of constant gradient $\tan\theta$.