How to graph polar roses and count their petals

A polar rose repeats a loop as its radius changes with the angle. The number multiplying the angle controls how many petals we get, and the outside coefficient controls their length. Let’s use both pieces of information before plotting a cosine rose.

Learn Precalculus with meStep-by-step videos, lesson notes, and practice for functions, polar curves, and more.

Explore Precalculus →

Read the coefficient and angle multiplier

For $r=a\cos \left(n\theta \right)$ or $r=a\sin \left(n\theta \right)$ with nonzero $a$ and positive integer $n$, each petal extends $\left|a\right|$ units from the pole. If $n$ is odd, there are $n$ petals. If $n$ is even, there are $2n$ petals.

Make a table for a four-petal rose

For $r=3\cos \left(2\theta \right)$, the petals have length $3$ and there are $4$ of them. The radius is zero when $\cos \left(2\theta \right)=0$, which gives the steps $\theta =\frac{\pi }{4},\frac{3\pi }{4},\dots$.

Sample the radius every pi over four

Angle	Radius
$0$	$3$
$\frac{\pi }{4}$	$0$
$\frac{\pi }{2}$	$-3$
$\frac{3\pi }{4}$	$0$
$\pi$	$3$
$\frac{5\pi }{4}$	$0$
$\frac{3\pi }{2}$	$-3$
$\frac{7\pi }{4}$	$0$

A negative radius is plotted opposite the angle’s ray. Following the table in order traces the four petals along the coordinate axes.

The four petal tips are (3, 0), (0, 3), (−3, 0), and (0, −3).

Apply the count to other roses

The curve $r=-5\cos \left(4\theta \right)$ has eight petals of length $5$. The curve $r=-6\sin \left(5\theta \right)$ has five petals of length $6$. Use sample angles and the sign of the radius to place the petals.

Full Precalculus course

Learn Precalculus with me

We’ll work through functions, polar curves, and other Precalculus topics, with the steps and examples explained as we go.

Explore my Precalculus course →

• Video lessons
• Lesson notes
• Practice problems

Included with membership, along with every other Krista King Math course.