Solution
= Solution
The maximum of <sine> is one. The allowed parameter values giving $\sin(3t)=1$ are
$$
t=\frac\pi6,\quad\frac{5\pi}6,\quad\frac{3\pi}2.
$$
Substituting each into $x=\cos t$ gives
$$
\boxed{y_{\max}=1,\qquad x\in\left\{\frac{\sqrt3}{2},-\frac{\sqrt3}{2},0\right\}.}
$$
All three parameter values lie in the original interval, so none of these maxima is discarded.