Problem 579

Solve
$y=5\left(\frac{3}{5} \sin x-\frac{4}{5} \cos x\right)=5 \sin (x-\varphi)$, when $y$ reaches its maximum value,
$$x_{0}-\varphi=\frac{\pi}{2} \Rightarrow \sin x_{0}=\cos \varphi=\frac{3}{5}, \cos x_{0}=-\sin \varphi=-\frac{4}{5} \Rightarrow \tan x_{0}=-\frac{3}{4}$$

6. $-\frac{3}{4}$.

Let $\cos \theta=\frac{3}{5}, \sin \theta=\frac{4}{5}$ ( $\theta$ is an acute angle).
Then $\tan \theta=\frac{4}{3}$.
Thus $y=5\left(\frac{3}{5} \sin x_{0}-\frac{4}{5} \cos x_{0}\right)$
$$=5 \sin \left(x_{0}-\theta\right) \leqslant 5 \text {. }$$

When $x_{0}-\theta=2 k \pi+\frac{\pi}{2}$, the equality holds, at this time,
$$\begin{array}{l} \tan x_{0}=\tan \left(2 k \pi+\frac{\pi}{2}+\theta\right)=\tan \left(\frac{\pi}{2}+\theta\right) \\ =-\cot \theta=-\frac{3}{4} . \end{array}$$