Question

How do you graph `r=4sin7theta`?

Answer 1

Graph is inserted. See explanation.

Explanation:

For using Socratic graphic utility, use the cartesian equivalent

graph{(x^2+y^2)^4=4(7x^6y-35x^4y^3+21x^2y^5-y^7) [-10, 10, -5, 5]}

Glory to De Moivre.

`sin 7theta` is periodic, with period `(2pi)/7`.

So, one rotation `theta in [0, 2pi]` creates 7 petals.

As `r = 4 sin 7theta >= 0 , 7theta in Q_1 or Q_2`, and this gives

`theta in [0, pi/7]`.

r becomes negative, in the other half of the period, `[pi/7, 2/7pi]`.

This happens for every loop.

The graphing algorithm ignores `r < 0`.

I take this opportunity to let others see the grandeur in the graph

of the counterpart `r = sin (theta/7)`:

graph{y-(x^2+y^2)(7+(x^2+y^2)(-56+(x^2+y^2)(112-64(x^2+y^2))))=0[-2 2 -1.2 1.2]}