Lesson 10.4 : Polar Coordinates: Graphs

New Concept

Ready to draw with math? This lesson introduces graphing polar equations. You'll learn how to test for symmetry and plot key points to sketch beautiful, classic curves like circles, cardioids, and roses from their equations.

What’s next

First, you'll learn how to test polar equations for symmetry. Then, you'll move on to interactive examples and graphing challenges to master each new curve.

Property

A polar equation describes a curve on the polar grid. The graph of a polar equation can be evaluated for three types of symmetry.

1. Symmetry with respect to the line $\theta = \frac{\pi}{2}$ ($y$-axis): Replace $(r, \theta)$ with $(-r, -\theta)$.
2. Symmetry with respect to the polar axis ($x$-axis): Replace $(r, \theta)$ with $(r, -\theta)$ or $(-r, \pi - \theta)$.
3. Symmetry with respect to the pole (origin): Replace $(r, \theta)$ with $(-r, \theta)$.

If the resulting equations are equivalent in one or more of the tests, the graph produces the expected symmetry. Passing a test verifies symmetry, but failing does not prove a lack of symmetry.

Examples

• Test $r = 5 \cos \theta$ for polar axis symmetry. Replacing $(r, \theta)$ with $(r, -\theta)$ gives $r = 5 \cos(-\theta)$, which simplifies to $r = 5 \cos \theta$. The equation is unchanged, so it is symmetric with respect to the polar axis.

• Test $r = 3 \sin \theta$ for symmetry about the line $\theta = \frac{\pi}{2}$. Replacing $(r, \theta)$ with $(-r, -\theta)$ gives $-r = 3 \sin(-\theta)$, which becomes $-r = -3 \sin \theta$, or $r = 3 \sin \theta$. It is symmetric with respect to the line $\theta = \frac{\pi}{2}$.

Property

To find the zeros of a polar equation, we solve for the values of $\theta$ that result in $r = 0$. Set $r = 0$, and solve for $\theta$. These are the points where the curve passes through the pole.
To find the maximum value of a polar equation, find the values of $\theta$ that result in the maximum value of the trigonometric functions. For example, for $r = a \cos \theta$, the maximum $|r|$ is $|a|$ because the maximum value of $\cos \theta$ is 1.

Examples

• Find the zeros of $r = 3 \cos \theta$. Set $r=0$, so $3 \cos \theta = 0$. This occurs when $\cos \theta = 0$, so $\theta = \frac{\pi}{2}$ and $\theta = \frac{3\pi}{2}$. The graph passes through the pole at these angles.

• Find the maximum value of $|r|$ for the equation $r = 4 + 3 \sin \theta$. The maximum value of $\sin \theta$ is 1. So, the maximum $r$ is $4 + 3(1) = 7$.

Property

Some of the formulas that produce the graph of a circle in polar coordinates are given by $r = a \cos \theta$ and $r = a \sin \theta$, where $a$ is the diameter of the circle or the distance from the pole to the farthest point on the circumference. The radius is $|\frac{a}{2}|$, or one-half the diameter. For $r = a \cos \theta$, the center is $(\frac{a}{2}, 0)$. For $r = a \sin \theta$, the center is $(0, \frac{a}{2})$.

Examples

• The graph of $r = 8 \cos \theta$ is a circle with diameter 8. Its radius is 4 and its center is at the point $(4, 0)$ on the polar axis.

• The graph of $r = 5 \sin \theta$ is a circle with diameter 5. Its radius is 2.5 and its center is at the point $(0, 2.5)$ on the line $\theta = \frac{\pi}{2}$.

Property

The formulas that produce the graphs of a cardioid or limaçon are given by $r = a \pm b \cos \theta$ and $r = a \pm b \sin \theta$ where $a > 0$, $b > 0$. The specific shape depends on the ratio $\frac{a}{b}$.

1. Cardioid: If $\frac{a}{b} = 1$, the graph is heart-shaped and passes through the pole.
2. One-Loop Limaçon: If $\frac{a}{b} > 1$, the graph is a single loop that does not pass through the pole. It may be dimpled ($1 < \frac{a}{b} < 2$) or convex ($\frac{a}{b} \ge 2$).
3. Inner-Loop Limaçon: If $\frac{a}{b} < 1$, the graph has an outer loop and a smaller inner loop, passing through the pole twice.

Examples

• The equation $r = 4 + 4 \cos \theta$ has $\frac{a}{b} = \frac{4}{4} = 1$. This graph is a cardioid.

• The equation $r = 5 - 3 \sin \theta$ has $\frac{a}{b} = \frac{5}{3} > 1$. This graph is a one-loop (dimpled) limaçon.

Property

The formulas that generate the graph of a lemniscate are given by $r^2 = a^2 \cos 2\theta$ and $r^2 = a^2 \sin 2\theta$ where $a \neq 0$. The formula $r^2 = a^2 \sin 2\theta$ is symmetric with respect to the pole. The formula $r^2 = a^2 \cos 2\theta$ is symmetric with respect to the pole, the line $\theta = \frac{\pi}{2}$, and the polar axis.

Examples

• The graph of $r^2 = 16 \cos 2\theta$ is a lemniscate. The maximum distance from the pole is $r = \sqrt{16} = 4$, which occurs at $\theta=0$ and $\theta=\pi$.

• The graph of $r^2 = 25 \sin 2\theta$ is a lemniscate oriented diagonally along the line $\theta = \frac{\pi}{4}$. The maximum distance from the pole is $r=5$.

Property

The formulas that generate the graph of a rose curve are given by $r = a \cos n\theta$ and $r = a \sin n\theta$ where $a \neq 0$. If $n$ is even, the curve has $2n$ petals. If $n$ is odd, the curve has $n$ petals.

Examples

• The graph of $r = 5 \cos 2\theta$ is a rose curve. Since $n=2$ (even), the curve has $2n = 4$ petals. The length of each petal is 5.

• The graph of $r = 4 \sin 3\theta$ is a rose curve. Since $n=3$ (odd), the curve has $n = 3$ petals. The length of each petal is 4.

Property

The formula that generates the graph of the Archimedes’ spiral is given by $r = \theta$ for $\theta \ge 0$. As $\theta$ increases, $r$ increases at a constant rate in an ever-widening, never-ending, spiraling path.

Examples

• For the spiral $r = \theta$, the point at $\theta = \frac{\pi}{2}$ is $(\frac{\pi}{2}, \frac{\pi}{2})$. The distance from the pole is approximately 1.57.

• For the spiral $r = \theta$, the point at $\theta = 2\pi$ is $(2\pi, 2\pi)$. After one full rotation, the distance from the pole is approximately 6.28.