1. The problem asks to find the x-values of all vertical asymptotes of the function $$y = \csc(5x)$$ in the interval $$[0, 2\pi)$$. 2. Recall that $$\csc(\theta) = \frac{1}{\sin(\theta)}$$, so vertical asymptotes occur where $$\sin(5x) = 0$$ because division by zero is undefined. 3. The sine function equals zero at integer multiples of $$\pi$$, so: $$\sin(5x) = 0 \implies 5x = n\pi, \quad n \in \mathbb{Z}$$ 4. Solve for $$x$$: $$x = \frac{n\pi}{5}$$ 5. We need all such $$x$$ in the interval $$[0, 2\pi)$$, so find all integers $$n$$ such that: $$0 \leq \frac{n\pi}{5} < 2\pi$$ 6. Multiply all parts by 5/$$\pi$$: $$0 \leq n < 10$$ 7. Therefore, $$n = 0, 1, 2, \ldots, 9$$. 8. The vertical asymptotes are at: $$x = 0, \frac{\pi}{5}, \frac{2\pi}{5}, \frac{3\pi}{5}, \frac{4\pi}{5}, \pi, \frac{6\pi}{5}, \frac{7\pi}{5}, \frac{8\pi}{5}, \frac{9\pi}{5}$$ Final answer: $$\boxed{x = \frac{n\pi}{5} \text{ for } n=0,1,2,\ldots,9}$$