Polar Co-ordinate System

Polar Co-ordinate System

Cartesian co-ordinates locate a point by how far it is across and up from the origin, $(x,y)$. Polar co-ordinates instead locate a point by its straight-line distance from a fixed point, and the direction of that line. Let O be the origin (called the pole) and OX the positive x-axis (called the polar axis). For any point P in the plane, let $OP=r$ and $\angle XOP=\theta$. The ordered pair $(r,\theta)$ then determines P's position exactly as $(x,y)$ does — $r$ tells us how far, and $\theta$ tells us in what direction.

Converting between the two systems. By exactly the same definition used throughout this chapter for a point at distance r from the origin, $x=r\cos\theta$ and $y=r\sin\theta$ — this is how to go from polar $(r,\theta)$ to Cartesian $(x,y)$. Going the other way, from Cartesian to polar, use the distance formula and the tangent ratio:

where θ must be chosen in whichever quadrant contains the point $(x,y)$, since $\tan\theta$ alone repeats every $\pi$ radians and cannot by itself distinguish, say, quadrant I from quadrant III.

Solved Example — polar co-ordinates of the Cartesian point (3, 3)

Step 1. Here $x=3,\ y=3$.

Step 2. $r=\sqrt{x^2+y^2}=\sqrt{3^2+3^2}=\sqrt{18}=3\sqrt2$. …