3.469 Given: $\cos \left(x-\frac{3 \pi}{2}\right)=-\frac{4}{5}, 0<x<\frac{\pi}{2}$. Find $A=\sin \frac{x}{2} \cdot \cos \frac{5 x}{2}$.
Solution. $A=\frac{-\sin 2 x+\sin 3 x}{2}=\frac{3 \sin x-4 \sin ^{3} x-2 \sin x \cdot \cos x}{2}$.

Since $\sin x=\frac{4}{5}, \cos x=\frac{3}{5}$, then $A=-\frac{38}{125}$.

Answer: $A=-\frac{38}{125}$.