We can compare to $\sum_{n=1}^{\infty} \frac{1}{n^2}$ because $|\sin(n)| \leq 1$, and since $\sum \frac{1}{n^2}$ converges, the original series converges

We need the Limit Comparison Test instead because sine is oscillating

The Direct Comparison Test is inconclusive because $\sin(n)$ changes sign

The series diverges because $\sin(n)$ is not always positive