Q.Find the polar co-ordinates of the point whose Cartesian co-ordinates are $(5, 5)$.

Cartesian coordinates locate a point P by how far it is horizontally and vertically from the origin, i.e. by (x, y). Polar coordinates locate the same point differently: by its straight-line distance r from a fixed point O (called the pole) and the angle θ that the ray OP makes with a fixed reference ray OX (called the polar axis). The ordered pair (r, θ) then fixes P's position exactly as (x, y) does. The two systems are linked through the very definitions used throughout this chapter: if OP = r and ∠XOP = θ, then by the coordinate definition of sine and cosine, $x = r\cos\theta$ and $y = r\sin\theta$. Reversing this — going from Cartesian to polar — uses the distance formula and the tangent ratio:

$r = \sqrt{x^2+y^2}, \qquad \tan\theta = \frac{y}{x}$ …