2018 AMC 12A Problem 9

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

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.