Question

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?

$y = 0$

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

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

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

$0 \le y \le \pi$

Answer: E

Concepts:trigonometric identity inequality

Difficulty rating: 1500

Solution:

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.

2018 AMC 12A Problem 9

Answer: E

Difficulty rating: 1500

Solution:

Problem 9 in Other Years