Question

Find the vertical asymptotes (if any) of the graph of the function. (Use n as an arbitrary integer if necessary. If an answer does not exist, enter DNE.) s(t) = 5t/sin(t)

Answer 1

$\bf s(t)=\cfrac{5t}{sin(t)}$

a rational expression is asymptotic, when the expression becomes undefined, and that happens when the denominator is 0, or turns to 0

the vertical asymptote is there, only if such value, turns the denominator to 0, whilst not zeroing out the numerator

well, for sine, that  is at $\bf -\infty...0,\pi , 2\pi ,3\pi ,4\pi ....+\infty \\\\\\ thus\implies 0\pm \pi n\quad \textit{"n" being an integer}$