Question

How do you graph `r^2 = sin2(t)`?

Answer 1

See graphs galore.

Explanation:

The power-scaling of r is done,

when `r` is changed to `r^n`, n > 1, in `r = f( theta )`.

It is r-more, when `r in ( 0, 1 )`. Otherwise, jt is r-less

Here, this is an example.

Use `r = sqrt( x^2 + y^2 ) >= 0, r ( cos theta, sin theta )`

and `sin 2theta = 2 sintheta cos theta`,

to get the Cartesian form of

`r^2 = sin 2theta` as

`( x^2 + y^2 ) ^2 ) - 2xy = 0`.

The Socratic graph is immediate.
graph{( x^2 + y^2 ) ^2 - 2xy =0[-2 2 -1 1]}
Graph of `r = sin 2theta`, for contrast in r-scaling:
graph{( x^2 + y^2 ) ^1.5 - 2xy =0[-2 2 -1 1]}

Easy to see which is which, jn the combined graph, along with the

third graph of `r^5 = sin 2theta`:

graph{(( x^2 + y^2 ) ^2 - 2xy)( ( x^2 + y^2 ) ^1.5 - 2xy) ( ( x^2 + y^2 ) ^3.5 - 2xy) =0[-2 2 -1 1]}