Question

How do you graph `r=4sin(5theta)`?

Answer 1

Please see below.

Explanation:

`r=asinntheta` is a typical rose curve though it looks more like daisy. If `n` is odd, number of petals are `n`, but if `n` is even, number of petals are `2n`. For example `r=3sin2theta` looks like

graph{((x^2+y^2)^(3/2)-6xy)((x^2+y^2)^(3/2)+6xy)=0 [-10, 10, -5, 5]}

But `r=4sin5theta` is

graph{(x^2+y^2)^3=20y(x^2+y^2)^2-80y^3(x^2+y^2)+64 y^5 [-10, 10, -5, 5]}

Observe that when `theta=pi/10` we have `r=4` and when `theta=pi/5`, `r=0`. Again at `theta=(3pi)/10` `r=-4` i.e. maximum on the opposite side and when `theta=(4pi)/10`, `r=0` and so on and thus forming `5` petals.