# How do you graph r=cos(theta/n) for n=2, 4, 6 and 8 and how do you predict the shap of the graph for r=cos(theta/10)?

Nov 26, 2016

I could make the graph in the Cartesian frame only, See details.

#### Explanation:

$r = \cos \left(\frac{\theta}{n}\right) \ge 0 \to \frac{\theta}{n} \in \left[- \frac{\pi}{2} , \frac{\pi}{2}\right] \to \theta \in \left[- \frac{n}{2} \pi , \frac{n}{2} \pi\right] , n = 2 , 4 , 6 , \ldots$

Using conversion to (x, y), I have inserted n = 2 graph,

for $r = \cos \left(\frac{\theta}{2}\right)$

I have also succeeded in making it, for n = 4, 5, 6, 7, 8, 9, 10, ....

The trend is clear, for the limiting case. As $n \to \infty , r \to \cos 0 = 1$,

representing a $u n i t$ circle. Also, for n = 1, the graph is a

$\frac{1}{2} u n i t$ circle

Graph for n = 1 Circle.
graph{(x^2+y^2)-x=0[-2 2 -1 1]}
Graph for n = 2 dimpled Cardioid.
graph{x-(x^2+y^2)^0.5(2(x^2+y^2)-1)=0[-2 2 -1 1]}
Graph for $n = 4$, with a large innie.
graph{x-(x^2+y^2)^0.5(1-8(x^2+y^2)(1-x^2-y^2))=0 [-2 2 -1.2 1.2]}

5-in-1 graph, for n = 1, 2, 4, 10 and $\infty$:
graph{(x-(x^2+y^2)^0.5((x^2+y^2)^5-(1-x^2-y^2)^5-45(x^2+y^2)(1-x^2-y^2)((x^2+y^2)^3-(1-x^2-y^2)^3)+210(x^2+y^2)^2(1-x^2-y^2)^2(2(x^2+y^2)-1)))(x-(x^2+y^2)^0.5(1-8(x^2+y^2)(1-x^2-y^2)))(x-(x^2+y^2)^0.5(2(x^2+y^2)-1))((x^2+y^2)^0.5-1)(x^2+y^2-x)=0 [-2 2.2 -1.2 1.2]}

Jun 3, 2018

My second answer is exclusive for the graph of $r = \cos \left(\frac{\theta}{10}\right)$, in Cartesian frame.

#### Explanation:

The period is $20 \pi$. For half period $10 \pi$ only $r \ge 0$.

Observe five $2 \pi$-rotations of tracing, before returning to the

start at (1, 0) that is fixed for all n. The opposite end moves

towards (-1, 0), for the limit.Slide the graph rarr darr larr

uarr, to see all sides of this zoomed graph, on uniform scale.

De Moivre's theorem is used.
graph{x-(x^2+y^2)^0.5((x^2+y^2)^5-(1-x^2-y^2)^5-45(x^2+y^2)(1-x^2-y^2)((x^2+y^2)^3-(1-x^2-y^2)^3)+210(x^2+y^2)^2(1-x^2-y^2)^2(2(x^2+y^2)-1))=0[-1 1 -0.5 0.5]}.

See also the 5-loop neighbor behind in the sequence, $r = \cos \left(\frac{\theta}{9}\right)$. Long pending issues are now settled.

graph{x -(x^2+y^2)^5+36(x^2+y^2)^4(1-x^2-y^2)-126(x^2+y^2)^3(1-x^2-y^2)^2+84(x^2+y^2)^2(1-x^2-y^2)^3-9(x^2+y^2)(1-x^2-y^2)^4=0[-2 2 -1 1]}