Find a Lipschitz constant w.r.t. $y$ of $f(x,y) = \sin(xy)$
Your method works, because $|x| \leq 3$ on this domain, so making that replacement gives a Lipschitz constant. However you actually discarded the division by 2 for no real reason so you ended up with a suboptimal constant.
A faster way to do it is to use a bound on the partial derivative:
$$\left | \frac{\partial f}{\partial y} \right | =\left | x\cos(xy) \right | \leq |x| \leq 3.$$
The fact that this is sufficient is a consequence of the mean value theorem.
Note that my result is the same as what you would get from your method if you didn't discard the division by 2.