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2018 AMC 12A Problem 9

Below is the professionally curated solution for Problem 9 of the 2018 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 12A solutions, or check the answer key.

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Concepts:trigonometric identityinequality

Difficulty rating: 1500

9.

Which of the following describes the largest subset of values of yy within the closed interval [0,π][0, \pi] for which sin⁡(x+y)≤sin⁡(x)+sin⁡(y) \sin(x + y) \le \sin(x) + \sin(y) for every xx between 00 and π,\pi, inclusive?

y=0y = 0

0≤y≤π40 \le y \le \tfrac{\pi}{4}

0≤y≤π20 \le y \le \tfrac{\pi}{2}

0≤y≤3π40 \le y \le \tfrac{3\pi}{4}

0≤y≤π0 \le y \le \pi

Solution:

For 0≤x≤π0 \le x \le \pi and 0≤y≤π0 \le y \le \pi we have sin⁡x≥0,\sin x \ge 0, sin⁡y≥0,\sin y \ge 0, cos⁡x≤1,\cos x \le 1, and cos⁡y≤1.\cos y \le 1. Hence sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y≤sin⁡x+sin⁡y. \sin(x+y) = \sin x \cos y + \cos x \sin y \le \sin x + \sin y. The inequality therefore holds for every yy with 0≤y≤π.0 \le y \le \pi.

Thus, the correct answer is E.

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