2018 AMC 12A Problems/Problem 9

Revision as of 15:12, 8 February 2018 by Benq (talk | contribs) (Created page with "== Problem == Which of the following describes the largest subset of values of <math>y</math> within the closed interval <math>[0,\pi]</math> for which <cmath>\sin(x+y)\leq \...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem

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

Solution

On the interval $[0, \pi]$ sine is nonnegative; thus $\sin(x + y) = \sin x \cos y + \sin y \cos x \le \sin x + \sin y$ for all $x, y \in [0, \pi]$. The answer is $\boxed{\textbf{(E) } 0\le y\le \pi}$. (CantonMathGuy)

See Also

2018 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Invalid username
Login to AoPS