Sample Question 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?

$\textbf{(A) } y=0 \qquad \textbf{(B) } 0\leq y\leq \frac{\pi}{4} \qquad \textbf{(C) } 0\leq y\leq \frac{\pi}{2} \qquad \textbf{(D) } 0\leq y\leq \frac{3\pi}{4} \qquad \textbf{(E) } 0\leq y\leq \pi$

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Answer Keys

Question 9: E

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Solutions

Question 9
Step 1: Consider the equation $\sin(x+y)\leq \sin(x)+\sin(y)$. In the interval [0, π], sine is nonnegative.

Step 2: We can rewrite the original equation using the addition formula for sine, which is $\sin(x + y) = \sin x \cos y + \sin y \cos x$.

Step 3: By doing so, we get $\sin x \cos y + \sin y \cos x \le \sin x + \sin y$.

Step 4: We can see that equality is obtained when $\cos x = \cos y = 1$, which is the maximum value of cosine. Therefore, the largest subset of values of $y$ for which the equation is satisfied is all $y$ in the interval [0, π].

Final Answer: $\textbf{(E) } 0\le y\le \pi$.