Polar equations and graphs

A polar equation relates $r$ and $\theta$, usually in the form $r = f(\theta)$. Curves that are messy in $x$ and $y$, like flowers, hearts and spirals, often have short, elegant polar equations. In this lesson you will graph the main families and convert equations between the two systems.

Graphing by plotting points

The graph of $r = f(\theta)$ is every point $(r, \theta)$ that satisfies the equation. The basic method is the same as for $y = f(x)$: make a table, plot, and connect. Think of sweeping a ray counterclockwise from $\theta = 0$ and watching how far out the point sits as the ray turns.

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

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

The point starts 4 units out on the positive $x$-axis, shrinks to the pole at $\theta = \pi$, then grows back. The result is a heart shape called a cardioid.

The standard families

You don't need a table every time. A few families come up again and again, and recognizing the form tells you the shape. In each family, $a$ and $b$ are positive constants.

Equation	Graph
$r = a$	circle of radius $a$ centered at the pole
$\theta = \alpha$	line through the pole at angle $\alpha$
$r = a\cos\theta$ or $r = a\sin\theta$	circle of diameter $a$ passing through the pole
$r = a \pm b\cos\theta$ or $r = a \pm b\sin\theta$	limaçon (cardioid when $a = b$)
$r = a\cos(n\theta)$ or $r = a\sin(n\theta)$	rose with petals of length $a$

Circles through the pole

The graph of $r = 4\sin\theta$ is a circle of diameter 4 sitting on top of the pole. The graph of $r = 4\cos\theta$ is the same circle turned to the right.

Limaçons

For $r = a + b\cos\theta$ (or with $\sin$), the ratio $\tfrac{a}{b}$ decides the shape:

• $a < b$: the curve has an inner loop, because $r$ becomes negative for some angles.
• $a = b$: a cardioid, touching the pole at one point.
• $a > b$: a dimpled or convex limaçon that never reaches the pole.

Roses

Why the difference? When $n$ is odd, the negative values of $r$ retrace petals that were already drawn. When $n$ is even, the negative values draw brand-new petals in the gaps.

Symmetry tests

Symmetry halves the plotting work. For a polar equation:

• Replacing $\theta$ with $-\theta$ gives an equivalent equation $\Rightarrow$ symmetric about the polar axis ($x$-axis). Every equation in $\cos\theta$ alone passes, since $\cos(-\theta) = \cos\theta$.
• Replacing $\theta$ with $\pi - \theta$ gives an equivalent equation $\Rightarrow$ symmetric about the line $\theta = \tfrac{\pi}{2}$ ($y$-axis). Equations in $\sin\theta$ alone pass, since $\sin(\pi - \theta) = \sin\theta$.
• Replacing $r$ with $-r$ gives an equivalent equation $\Rightarrow$ symmetric about the pole.

These tests are sufficient but not necessary: a curve can be symmetric even when a test fails.

Converting between polar and rectangular equations

Use the same tools as for points: $x = r\cos\theta$, $y = r\sin\theta$, $r^2 = x^2 + y^2$. The trick is often to multiply by $r$ so those combinations appear.

Intersections of polar curves

To find where two polar curves meet, set the $r$-expressions equal and solve for $\theta$. Then also check the pole separately: two curves can both pass through the pole at different angles, and setting them equal will miss it.

Practice

Practice 1

What is the graph of $r = 5$?

AA line through the pole at angle 5BA circle of radius 5 centered at the poleCA circle of diameter 5 passing through the poleDA rose with 5 petals

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Practice 2

How many petals does the rose $r = 2\sin(4\theta)$ have?

Your answer

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

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Practice 3

The polar equation $r = 6\sin\theta$ is a circle. Convert it to rectangular form and give the center as $(x, y)$.

Your answer

Enter a point like (2, -3)

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Practice 4

Which polar equation describes the vertical line $x = 4$?

A$r = 4$B$r = 4\cos\theta$C$r = 4\sin\theta$D$r = 4\sec\theta$

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Practice 5

The limaçon $r = 1 + 2\sin\theta$ passes through the pole. Find every $\theta$ in $[0, 2\pi)$ where $r = 0$.

Your answer

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

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Practice 6

Convert $r = \dfrac{3}{\cos\theta + \sin\theta}$ to a rectangular equation. Solve for $y$.

Your answer

Enter an expression, e.g. 3x^2 - 2x + 1

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Practice 7

The circles $r = 2$ and $r = 4\cos\theta$ are graphed below. Find every $\theta$ in $[0, 2\pi)$ where they meet, that is, where $4\cos\theta = 2$.

Your answer

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

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Practice 8

Which statement about $r = 3 - 3\cos\theta$ is true?

AIt is a cardioid symmetric about the polar axis, farthest from the pole at $\theta = \pi$.BIt is a limaçon with an inner loop, symmetric about the line $\theta = \tfrac{\pi}{2}$.CIt is a cardioid symmetric about the line $\theta = \tfrac{\pi}{2}$, farthest from the pole at $\theta = 0$.DIt is a rose with 3 petals.

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