Question

How do you create a graph of `r = sin ((4theta)/3)`?

Answer 1

See the 4-loop graph and the crisscrossing of loops. Also, see the idiosyncratic 5-loop graph of `r =sin ((5/4)theta)`.

Explanation:

Use astutely

`( x, y ) = r ( cos theta, sin theta )`,

`r = sqrt(x^2 + y^2) = sin ((4theta)/3)`

`sin 4theta = 4sin theta cos theta(cos^2theta - sin^2theta )`

`= sin (3(4/3)theta)`

`= 3 cos^2((4/3)theta) sin ((4/3)theta) - sin^3((4/3)theta)`

and arrive at the Cartesian form

`4xy(x^2-y^2) = (x^2+y^2)^2.5(3-4(x^2+y^2))`.

The Socratic graph:.

graph{4xy(x^2-y^2)-(x^2+y^2)^2.5(3-4(x^2+y^2))=0[-2 2 -1 1]}

Similar astute approach gives the graphs of `r = cos ((3/4)theta)` and # r = sin ((5/4)theta).

Graph of `r = cos ((3/4)theta)`:
graph{x^3-3xy^2-(x^2+y^2)^1.5(8(x^2+y^2)^2-8(x^2+y^2)+1)=0[-2 2 -1 1]}
Idiosyncratic graph of `r = sin ((5/4)theta)`:
graph{5x^4y-10x^2y^3+y^5-4(x^2+y^2)^3(1-x^2-y^2)^0.5(1-2(x^2+y^2))=0[-4 4 -2 2]}