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

Jul 23, 2018

See graphs galore.

#### Explanation:

The power-scaling of r is done,

when $r$ is changed to ${r}^{n}$, n > 1, in $r = f \left(\theta\right)$.

It is r-more, when $r \in \left(0 , 1\right)$. Otherwise, jt is r-less

Here, this is an example.

Use $r = \sqrt{{x}^{2} + {y}^{2}} \ge 0 , r \left(\cos \theta , \sin \theta\right)$

and $\sin 2 \theta = 2 \sin \theta \cos \theta$,

to get the Cartesian form of

${r}^{2} = \sin 2 \theta$ 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 $r = \sin 2 \theta$, 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 ${r}^{5} = \sin 2 \theta$:

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]}