Polar Coordinates and Graphs of Polar Functions

Two dictionaries

From polar to rectangular: $x=r\cos\theta$, $y=r\sin\theta$. From rectangular to polar: $r^2=x^2+y^2$, $\tan\theta=\frac{y}{x}$ (choose the quadrant by looking at the signs of $x$ and $y$, never blindly from arctan).

A polar point has infinitely many names: $(r,\theta)=(r,\theta+2\pi k)=(-r,\theta+\pi)$. Negative $r$ means "walk backwards": plot the angle, then step $|r|$ in the opposite direction.

The standard graphs

• $r=a\cos\theta$: circle of diameter $|a|$ through the pole, centered on the $x$-axis; $r=a\sin\theta$ centers on the $y$-axis.
• $r=a\pm b\cos\theta$ (or $\sin$): limaçon — cardioid when $a=b$; inner loop when $a
• $r=a\cos(n\theta)$ or $a\sin(n\theta)$: rose with $n$ petals if $n$ is odd, $2n$ petals if $n$ is even, petal length $|a|$.

Rates of change

$r$ is a function of $\theta$, so "increasing/decreasing" questions are about $r(\theta)$, and the average rate of change on $[\theta_1,\theta_2]$ is $\frac{r(\theta_2)-r(\theta_1)}{\theta_2-\theta_1}$. Careful: the distance to the pole is $|r|$. While $r$ is negative and decreasing, the point is actually moving away from the pole.