Concept preview

What you will be able to do

1. 01Locate and plot points in polar coordinates $(r,\theta)$ and convert between polar and rectangular form using $x=r\cos\theta$ and $y=r\sin\theta$.
2. 02Interpret a polar curve $r=f(\theta)$ and recognise the shapes of circles, roses, and cardioids directly from their equations.
3. 03Differentiate a polar curve by treating it as a parametric curve in $\theta$, using $\dfrac{dy}{dx}=\dfrac{(dr/d\theta)\sin\theta+r\cos\theta}{(dr/d\theta)\cos\theta-r\sin\theta}$, and locate horizontal and vertical tangents.

A polar coordinate pair $(r,\theta)$ names a point by two instructions: face in the direction of angle $\theta$ (measured counter-clockwise from the positive $x$-axis), then walk a distance $r$ outward from the origin, called the pole. The point $(2,\,\pi/3)$, for instance, sits two units out along the ray that makes a $60^\circ$ angle with the positive $x$-axis. Angles are in radians throughout calculus, and a negative $r$ means "walk backwards" — the point $(-2,\,\pi/3)$ lands in the opposite direction, the same place as $(2,\,\pi/3+\pi)$.