How do you graph #r=4sin7theta#?

1 Answer
Jan 13, 2017

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]}