Precalculus / Unit 7: Polar Coordinates and Complex Numbers
7.3 Basic Polar Graphs
Notes
- Circles: $r = a\cos\theta$ or $r = a\sin\theta$.
- Limacons: $r = a + b\cos\theta$ or $r = a + b\sin\theta$. If $a = b$ the graph is a cardioid (heart shape).
- Rose curves: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$ have $n$ petals if $n$ is odd and $2n$ petals if $n$ is even.
- The maximum value of $|r|$ often gives the farthest point from the origin. For $r = 2 + 2\cos\theta$ it is 4.
Practice
How many petals does $r = 3\cos 2\theta$ have?
When $n$ is even there are $2n$ petals.
How many petals does $r = 2\sin 5\theta$ have?
When $n$ is odd there are $n$ petals.
What shape is the graph of $r = 3 + 3\sin\theta$?
When $a = b$ a limacon is a cardioid.
Ready for more? Take the Unit 7 test.