Math Core

In rectangular coordinates, $y = f(x)$ tells you how high the graph is above each $x$ . A polar equation $r = f(\theta)$ tells you how far the graph is from the pole in each direction. As $\theta$ sweeps around the circle, $r$ grows and shrinks, and the point traces out shapes that are hard to write in $x$ and $y$ : hearts, loops and flowers.

Graphing by plotting points

The most reliable method is to make a table. Choose angles around the circle, compute $r$ , and plot each point by turning to the angle and walking out the distance $r$ .

Worked example: A cardioid, point by point

Graph $r = 2 + 2\cos\theta$ .

$\theta$	$0$	$\tfrac{\pi}{3}$	$\tfrac{\pi}{2}$	$\tfrac{2\pi}{3}$	$\pi$	$\tfrac{4\pi}{3}$	$\tfrac{3\pi}{2}$	$\tfrac{5\pi}{3}$
$r$	$4$	$3$	$2$	$1$	$0$	$1$	$2$	$3$

At $\theta = 0$ the point is $4$ units to the right. As $\theta$ increases to $\pi$ , $r$ shrinks to $0$ , so the curve curls into the pole. Then it grows again on the way back around. The result is a heart shape called a cardioid.

The cardioid r = 2 + 2cos(θ). It reaches r = 4 at θ = 0 and touches the pole at θ = π.Open in grapher →

Symmetry

A table goes faster if you know the graph is symmetric, because then you only need half of it.

Symmetry tests for polar graphs

Replacing $\theta$ with $-\theta$ gives an equivalent equation: the graph is symmetric about the polar axis (the $x$ -axis).
Replacing $\theta$ with $\pi - \theta$ gives an equivalent equation: the graph is symmetric about the line $\theta = \tfrac{\pi}{2}$ (the $y$ -axis).
Replacing $r$ with $-r$ gives an equivalent equation: the graph is symmetric about the pole.

Since $\cos(-\theta) = \cos\theta$ , equations built from $\cos\theta$ are symmetric about the polar axis. Since $\sin(\pi - \theta) = \sin\theta$ , equations built from $\sin\theta$ are symmetric about the $y$ -axis. These tests are sufficient but not necessary: a graph can have a symmetry that the test fails to detect.

A catalog of polar graphs

A few families show up again and again. Knowing their shapes lets you sketch them quickly and check your tables.

Circles and lines. $r = a$ is a circle of radius $|a|$ centered at the pole. $\theta = \alpha$ is a line through the pole. $r = a\cos\theta$ and $r = a\sin\theta$ are circles of diameter $|a|$ that pass through the pole, centered on the $x$ -axis and $y$ -axis respectively.

Limaçons. Equations of the form $r = a + b\cos\theta$ or $r = a + b\sin\theta$ (with $a, b > 0$ ) are called limaçons. Their shape depends on the ratio $\dfrac{a}{b}$ :

ratio	shape
$\dfrac{a}{b} < 1$	inner loop
$\dfrac{a}{b} = 1$	cardioid (touches the pole)
$1 < \dfrac{a}{b} < 2$	dimpled
$\dfrac{a}{b} \ge 2$	convex (no dent)

The same shapes appear with a minus sign, just flipped: $r = a - b\cos\theta$ points left instead of right.

Two limaçons: r = 1 + 2cos(θ) has an inner loop, and r = 3 + 2cos(θ) is dimpled.Open in grapher →

Rose curves. $r = a\cos(n\theta)$ and $r = a\sin(n\theta)$ , with $n$ a positive integer of at least $2$ , are roses. Each petal has length $|a|$ .

Counting petals

For $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$ :

if $n$ is odd, the rose has $n$ petals;
if $n$ is even, the rose has $2n$ petals.

r = 4cos(3θ) has 3 petals of length 4. r = 3sin(2θ) has 4 petals of length 3.Open in grapher →

Why the odd/even rule? When $n$ is odd, the petals traced with negative $r$ land right on top of petals already drawn, so you only see $n$ . When $n$ is even, they land in new spots, doubling the count.

Common mistake

Don't read $n$ petals for every rose. $r = 2\sin(4\theta)$ has $8$ petals, not $4$ . Check whether $n$ is odd or even first.

Finding maximum $r$ and zeros

Two quick questions give you the skeleton of any polar graph: where is it farthest from the pole, and where does it pass through the pole?

Worked example: Analyzing a limaçon

For $r = 1 - 2\sin\theta$ , find the maximum value of $|r|$ and every $\theta$ in $[0, 2\pi)$ where the graph passes through the pole.

Maximum. $\sin\theta$ ranges from $-1$ to $1$ , so $r$ ranges from $1 - 2 = -1$ to $1 + 2 = 3$ . The largest distance is $|r| = 3$ , at $\theta = \tfrac{3\pi}{2}$ , where $\sin\theta = -1$ .

Zeros. Set $r = 0$ : $1 - 2\sin\theta = 0$ , so $\sin\theta = \tfrac{1}{2}$ and $\theta = \tfrac{\pi}{6}$ or $\tfrac{5\pi}{6}$ .

Since $\dfrac{a}{b} = \dfrac{1}{2} < 1$ , the graph is a limaçon with an inner loop. The two zeros are where the loop crosses the pole.

Worked example: Analyzing a rose

Describe $r = 5\cos(2\theta)$ .

$n = 2$ is even, so there are $2 \cdot 2 = 4$ petals, each of length $5$ . The tips are where $|\cos 2\theta| = 1$ , which is at $\theta = 0, \tfrac{\pi}{2}, \pi, \tfrac{3\pi}{2}$ . So the petals point along the axes.

Tip

A graphing tool is a great check, but know the family first. If your grapher shows $3$ petals for $r = 2\cos(4\theta)$ , you've typed something wrong.

Practice

Practice 1

How many petals does the graph of $r = 5\sin(4\theta)$ have?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

How long is each petal of the rose $r = 4\cos(3\theta)$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

What is the maximum value of $r$ for $r = 3 + 2\sin\theta$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

What kind of graph is $r = 2 - 3\cos\theta$ ?

Practice 5

The graph of $r = 2 + 2\cos\theta$ is symmetric about which line?

Practice 6

The graph of $r = 10\sin\theta$ is a circle. Find its center in rectangular coordinates.

Enter a point like (2, -3)

Practice 7

Find every $\theta$ in $[0, 2\pi)$ where the graph of $r = 1 + 2\cos\theta$ passes through the pole. Give your answers in radians, separated by commas.

Separate answers with commas, e.g. 2, -5

Practice 8

The rose $r = 3\cos(5\theta)$ and the rose $r = 3\cos(10\theta)$ are drawn. How many times as many petals does the second have as the first?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Graphing by plotting points

Symmetry

A catalog of polar graphs

Finding maximum rrr and zeros

Practice

Finding maximum $r$ and zeros