Question

How do you graph the polar equation `r=-2sin3theta`?

Answer 1

See graphs of `r =+- 2 sin 3theta`, to know how to get one from the other, using
`r = 2 sin 3(theta + alpha)`, for anticlockwise rotation through `alpha`.

Explanation:

See the anticlockwise rotation, about pole, of the graph of

`r = 2 sin 3theta`, giving the graph of `r = -2 sin 3theta`

`r = - 2 sin 3theta = 2 sin (pi + 3theta) = 2 sin (3(theta + pi/3))`

Formula for

anticlockwise rotation through `alpha` of `r = f ( theta )`::

`r - f ( theta + alpha )`

Graph of `r = - 2 sin 3theta`:

graph{(x^2+y^2)^2+2(3x^2y-y^3)=0}}

Graph of `r = 2 sin 3theta` :
graph{(x^2+y^2)^2-2(3x^2y-y^3)=0}}

For rotation through `+-pi/2`, see graphs of `r = +- 2 cos 3theta`:
Graph of `r = - 2 cos 3theta`:
graph{(x^2+y^2)^2-2(x^3-3xy^2)=0}

Graph of `r = 2 cos 3theta`:
graph{(x^2+y^2)^2+2(x^3-3xy^2)=0}