A rose curve has the equation $r=5\sin2\theta$. By solving appropriate equations, algebraically determine the angles of rotation on which the tips of petals will occur for $\theta\in\Bbb R$.
I have no idea what they mean by the angles of rotation (and thus I don't know how to solve this either) - the graph is given already, isn't it?
"Angles of rotation" is a little verbose. They mean "arguments of the petals' tips" and this entails solving for where $5\sin2\theta$ attains its extrema.
Since $-1\le\sin x\le1$ for real $x$, we only need to solve $\sin2\theta=1$ and $\sin2\theta=-1$, the solutions with $\theta\in[0,2\pi)$ being $\theta=\pi/4\lor\theta=5\pi/4$ and $\theta=7\pi/4\lor\theta=3\pi/4$ respectively. Thus the general solution for the $\theta$ at which petal tips occur is $\theta=(2n+1)\pi/4$, $n\in\Bbb Z$.