Which of the following describes the largest subset of values of $y$ within the closed interval $[0,\pi]$ for which\[\sin(x+y)\leq \sin(x)+\sin(y)\]for every $x$ between $0$ and $\pi$, inclusive? $\textbf{(A) } y=0 \qquad$ $ \textbf{(B) } 0\leq y\leq \frac{\pi}{4} \qquad$ $ \textbf{(C) } 0\leq y\leq \frac{\pi}{2} \qquad$ $ \textbf{(D) } 0\leq y\leq \frac{3\pi}{4} \qquad$ $ \textbf{(E) } 0\leq y\leq \pi$