Polar coordinates, conversion between polar and Cartesian form, and sketching polar curves r = f(theta) including circles, lines, cardioids and spirals.

What this dot point is asking

OCR wants you to represent a point in polar coordinates $(r, \theta)$, convert between polar and Cartesian coordinates using $x = r\cos\theta$ and $y = r\sin\theta$, and sketch polar curves $r = f(\theta)$, identifying key features such as where the curve passes through the pole, its maximum $r$, and its symmetry, for standard curves including circles, lines, cardioids and spirals.

Polar coordinates of a point

In polar coordinates a point is located by how far it is from the pole and in what direction, rather than by horizontal and vertical displacement. The angle $\theta$ is measured anticlockwise from the initial line.

Converting between polar and Cartesian

The conversion formulae come straight from right-angled triangle trigonometry, with the point at distance $r$ and angle $\theta$.

Converting a polar equation

To turn a polar equation into Cartesian form, look to create the combinations $r^2$, $r\cos\theta$ and $r\sin\theta$, which become $x^2 + y^2$, $x$ and $y$. Multiplying both sides by $r$ is the standard trick when the equation has a lone $r$ or a $\cos\theta$ to be paired with one.

Sketching polar curves

To sketch $r = f(\theta)$, build a small table of $r$ at $\theta = 0, \tfrac{\pi}{4}, \tfrac{\pi}{2}, \ldots$, note where $r = 0$ (the curve passes through the pole, often giving a tangent direction) and where $r$ reaches its maximum, and use any symmetry (a curve in $\cos\theta$ is symmetric about the initial line, while one in $\sin\theta$ is symmetric about the line $\theta = \tfrac{\pi}{2}$). Recognise the standard shapes: $r = a$ (circle radius $a$), $\theta = \alpha$ (half-line), $r = a(1 + \cos\theta)$ (cardioid, a heart shape), $r = a\cos\theta$ (a circle through the pole), and $r = a\theta$ (an Archimedean spiral that widens steadily as $\theta$ increases). Plotting just a handful of well-chosen points, together with the zeros and maxima of $r$, is usually enough to produce an accurate sketch.

Polar coordinates set up the area formula for polar curves and connect to the modulus-argument form of complex numbers, where $r$ and $\theta$ play the same roles.

Try this

Q1. Convert $r = 3$ to Cartesian form. [1 mark]

• Cue. $r^2 = 9$ gives $x^2 + y^2 = 9$, a circle of radius $3$.

Q2. Find where the cardioid $r = 2(1 - \cos\theta)$ passes through the pole. [2 marks]

• Cue. $r = 0$ when $\cos\theta = 1$, that is $\theta = 0$.