Question

How do you graph `r=-8sin3theta`?

Answer 1

It has a 3 blade propeller shape.

Explanation:

I tried graphing `r = -8sin(3theta)` in both polar and Cartesian coordinates; neither is easy.

I used WolframAlpha to convert to functions of `theta`:

`r = 8sin^3(theta) - 24sin(theta)cos^2(theta)`

Multiply both sides by `r^3`:

`r^4 = 8r^3sin^3(theta) - 24rsin(theta)r^2cos^2(theta)`

Applying the polar to rectangular conversion equations:

`(x^2 + y^2)^2 = 8y^2 - 24y(x^2)`

Here is the graph:

![Desmos.com]()