Difference between revisions of "2018 AMC 12A Problems/Problem 9"
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− | On the interval <math>[0, \pi]</math> sine is nonnegative; thus <math>\sin(x + y) = \sin x \cos y + \sin y \cos x \le \sin x + \sin y</math> for all <math>x, y \in [0, \pi]</math>. The answer is <math>\boxed{\textbf{(E) } 0\le y\le \pi}</math>. (CantonMathGuy) | + | On the interval <math>[0, \pi]</math> sine is nonnegative; thus <math>\sin(x + y) = \sin x \cos y + \sin y \cos x \le \sin x + \sin y</math> for all <math>x, y \in [0, \pi]</math> and equality only occurs when <math>\cos x = \cos y = 1</math>, which is cosine's maximum value. The answer is <math>\boxed{\textbf{(E) } 0\le y\le \pi}</math>. (CantonMathGuy) |
==Solution 2== | ==Solution 2== |
Revision as of 20:20, 21 January 2020
Contents
[hide]Problem
Which of the following describes the largest subset of values of within the closed interval
for which
for every
between
and
, inclusive?
Solution 1
On the interval sine is nonnegative; thus
for all
and equality only occurs when
, which is cosine's maximum value. The answer is
. (CantonMathGuy)
Solution 2
Expanding, Let
,
. We have that
Comparing coefficients of
and
gives a clear solution: both
and
are less than or equal to one, so the coefficients of
and
on the left are less than on the right. Since
, that means that this equality is always satisfied over this interval, or
.
Solution 3
If we plug in , we can see that
. Note that since
is always nonnegative,
is always nonpositive. So, the inequality holds true when
. The only interval that contains
in the answer choices is
.
See Also
2018 AMC 12A (Problems • Answer Key • Resources) | |
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 |
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