It's better to avoid using x to mean multiply. 6sin(96) is easy to understand. Let $\displaystyle \theta$ be the angle you need to find
$\displaystyle \sin \theta = \dfrac{6\sin(96)}{10}$
Never round off halfway through a question, especially to one decimal place!
$\displaystyle \theta = \arcsin \left( \dfrac{6\sin(96)}{10}\right) = 36.6^{\circ} \text{ 3sf }$
You've got the method, you just need to look at your rounding
Your approach is certainly correct (law of sines). I get a very slightly different answer:
$\displaystyle \dfrac{\sin(?)}{6}=\dfrac{\sin(96^{\circ})}{10}$
$\displaystyle \sin(?)=0.6\,\sin(96^{\circ})$
$\displaystyle ?=\sin^{-1}(0.6\,\sin(96^{\circ}))\approx 36.63^{\circ}.$