Yes, I understand that. It makes perfect sense for $m,~n,~p~\&~q$.

But it makes no sense for the rest of the question. $\angle BAD$ is not a right angle.

Agree, however, $\angle BDC$ is a right angle.

The structure of the problem was strange in that it implied knowing $\sin(\angle ABD)$ leads directly toward determining the length of BC.

I had to find $\cos(\angle DBC)$ (which it didn't ask for) to determine the length of BC. Once that was determined it was rather straightforward to finish.

Someone may have a better way ...