Which of the following describes the largest subset of values of y within the closed interval [0, π] for which sin(x + y) ≤ sin(x) + sin(y) for every x between 0 and π, inclusive?

2018 AMC 12A Problem 9

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Which of the following describes the largest subset of values of

y

within the closed interval

[0, \pi]

for which

\sin(x + y) \le \sin(x) + \sin(y)

for every

x

between

0

and

\pi,

inclusive?

y = 0

0 \le y \le \tfrac{\pi}{4}

0 \le y \le \tfrac{\pi}{2}

0 \le y \le \tfrac{3\pi}{4}

0 \le y \le \pi

Expand

\sin(x+y)

= \sin x \cos y + \cos x \sin y

For

x, y \in [0, \pi]

both

\sin x, \sin y \ge 0

and

\cos x, \cos y \le 1;

compare term by term

For

0 \le x \le \pi

and

0 \le y \le \pi

we have

\sin x \ge 0,

\sin y \ge 0,

\cos x \le 1,

and

\cos y \le 1.

Hence

\begin{aligned} \sin(x+y) &= \sin x \cos y \\ &\quad {}+ \cos x \sin y \\ &\le \sin x + \sin y. \end{aligned}

The inequality therefore holds for every

y

with

0 \le y \le \pi.

Thus, the correct answer is E.