A polar equation $r = f(\theta)$ gives a distance for each direction. As $\theta$ sweeps around, the point traces a curve: circles, hearts, loops and flowers that would be hard to describe with $x$ and $y$. Plot them with polar coordinates, one direction at a time.
Key idea
Circles: $r = a$; $r = a\cos\theta$ and $r = a\sin\theta$ (through the pole). Limaçons: $r = a \pm b\cos\theta$ (or $\sin\theta$) with $a, b > 0$: a cardioid when $a = b$, an inner loop when $a < b$, no loop when $a > b$. Roses: $r = a\cos(n\theta)$ or $a\sin(n\theta)$ with $n \ge 2$: $n$ petals if $n$ is odd, $2n$ if $n$ is even, each of length $|a|$.
Why it works
For each angle, $r$ says how far out along that ray the curve is. When $r$ shrinks to $0$, the curve passes through the pole; when $r$ turns negative, the point jumps to the opposite side of the pole. That is what draws the inner loop of a limaçon and the extra petals of a rose. For odd $n$, the negative half of a rose retraces petals already drawn, so only $n$ appear.
Worked examples
Worked example: A cardioid
Sketch $r = 2 + 2\cos\theta$.
- Table: $\theta = 0$ gives $r = 4$; $\frac{\pi}{2}$ gives $2$; $\pi$ gives $0$; $\frac{3\pi}{2}$ gives $2$.
- Replacing $\theta$ with $-\theta$ leaves $\cos\theta$ unchanged, so the curve is symmetric about the polar axis.
- Join the points: the curve reaches out $4$ units to the right and comes into the pole on the left.
Answer: A cardioid (heart shape) with its point at the pole
Worked example: A rose
Describe $r = 3\sin 2\theta$.
- $n = 2$ is even, so there are $2n = 4$ petals, each $3$ units long.
- $|r|$ is largest where $\sin 2\theta = \pm 1$: $\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$, one petal centered on each of those rays.
Answer: A four-petal rose with petals along the diagonals
Worked example: An inner loop
Where does $r = 1 + 2\cos\theta$ pass through the pole?
- $r = 0$ when $\cos\theta = -\frac{1}{2}$: $\theta = \frac{2\pi}{3}$ and $\frac{4\pi}{3}$.
- Between those angles $r$ is negative, and that stretch of the curve forms a small loop inside the big one.
Answer: At $\theta = \frac{2\pi}{3}$ and $\frac{4\pi}{3}$
Common mistake
Plotting $r$ as a $y$-value against $\theta$ on ordinary axes. That graph is useful (it is the lower picture above), but it is not the polar curve. In a polar graph, every value of $\theta$ is a direction from the pole, so the curve wraps around the pole.