I need some help with a determinants problem

Question

I've tried to think about it but I don't know how to start

Answer 1

I would assume that the book has a typo, as I clarify in my comment above. In particular, I think that the question is really asking about $$ \begin{vmatrix} a&b&c\\ 5&7&8\\ \sin A & \sin B & \sin C \end{vmatrix} $$ where $a,b,c$ are the side lengths opposite the angles $A,B,C$. Assuming that this is the case,

Hint: note that, by the law of sines, we have $$ \frac{\sin(C)}{c} = \frac{\sin(B)}{b} \implies b\sin(C) = c \sin(B) $$ Using this inequality and others like it, take a Laplace expansion along the middle row to get the answer (d).

Or, even easier: because $\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$, we may conclude that the third row is a multiple of the first.