Answer
The Cartesian form of the expression is $\frac{1}{2x}=y$, which is a rotated hyperbola.
Work Step by Step
We are given that $r^{2}sin2\theta=1$.
Using the double-angle identity for sine, we can rewrite the expression as $r^{2}(2sin\theta cos\theta)=1$.
We can rewrite this expression as $2(rsin\theta)(rcos\theta)=1$.
Using the polar-to-cartesian substitutions of $x=rcos\theta$ and $y=rsin\theta$, we get $2xy=1$.
Solving for y, we get $\frac{1}{2x}=y$ and we can see that the graph is that of a scaled reciprocal function, which is a rotated hyperbola.
