Answer
$$x=\frac{3\pi}{10}+ \frac{ 2k\pi}{5}$$
Work Step by Step
Given $$ \sin^3 5x = -1 $$
Then
$$ \sin 5x =\sqrt[3]{-1} =-1 $$
Hence
$$ 5x=\frac{3\pi}{2}+2k\pi \ \Rightarrow x=\frac{3\pi}{10}+ \frac{ 2k\pi}{5}$$
where $k$ is any integer .
