AMC 10 · 2018 · #9

Grade 11 algebra
Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In
angle-addition-formulatrigonometric-ratiossign-analysisinterval-arithmetic complementary-countingsign-analysiseasier-related-problem ↑ Prerequisites: trigonometric-ratiosangle-addition-formula
📏 Medium solution 💡 3 insights
Problem
Find the largest set of y values inside the interval from zero to pi with this property: no matter which x is picked from that same interval, sine of x plus y is at most sine x plus sine y.

Pick an answer.

(A)
y=0
(B)
$0\leq y\leq \frac{\pi}{4}$
(C)
$0\leq y\leq \frac{\pi}{2}$
(D)
$0\leq y\leq \frac{3\pi}{4}$
(E)
$0\leq y\leq \pi$
How to solve
Strategy Organize Information in More Ways

The inequality as printed mixes x and y inside one sine, so nothing about it is readable. Tool #15 (Organize Information in More Ways): expand sin(x+y) with the angle addition formula, then move every term to one side and regroup so the expression becomes a sum of products whose signs can be read off one factor at a time. That regrouping is the whole problem. Tool #16 (Change Focus): instead of hunting for the y that work, ask what it would take for the inequality to fail — that question has a much shorter answer. Tool #14 (Extreme Principle): the sign facts that decide everything come from the extremes of sine and cosine on [0, π], namely sin ≥ 0 and cos ≤ 1. Tool #3 (Eliminate Possibilities): the choices are nested intervals, so there is only one threshold to find, and locating it picks the choice.

1STEP 1

Read the choices as one threshold

The choices are nested, so one test suffices.

{0} ⊂ [0, π/4] ⊂ [0, π/2] ⊂ [0, 3π/4] ⊂ [0, π]
2STEP 2

Ask how it could fail instead

Ask instead how it could fail.

x, y ∈ [0, π] → sin(x) + sin(y) ≥ 0; x + y ＞ π → sin(x+y) ≤ 0
3STEP 3

Expand the compound angle

Expand with the addition formula.

sin(x+y) = sin x cos y + cos x sin y
4STEP 4

Move everything to one side and regroup

Move everything over and group into two products.

sin x cos y + cos x sin y - sin x - sin y = sin x (cos y - 1) + sin y (cos x - 1)
5STEP 5

Pin down the sign of each factor

Each factor's sign is fixed on the interval.

sin x ≥ 0, sin y ≥ 0 on [0, π]; cos x - 1 ≤ 0, cos y - 1 ≤ 0 always
6STEP 6

Both products are nonpositive

Both products are at most zero.

sin x_ ≥ 0(cos y - 1)_ ≤ 0 + sin y_ ≥ 0(cos x - 1)_ ≤ 0 ≤ 0
7STEP 7

Check the boundary and the equality case

The boundary and equality cases pass too.

y = π: -sin x ≤ sin x; x = y = 3π/4: -1 ≤ √(2) ≈ 1.414
8STEP 8

Name the largest subset

So the answer is the whole interval.

{ y ∈ [0, π] : sin(x+y) ≤ sin x + sin y for all x ∈ [0, π] } = [0, π] → (E)
Answer
0 ≤ y ≤ π
The result says the condition never fails, which deserves a second look since four of the five choices assume it does. Two independent checks agree. First, the algebra: sin(x+y) - sin x - sin y = sin x(cos y - 1) + sin y(cos x - 1) is an identity, and on [0, π] each product is (nonnegative)×(nonpositive), so the difference is at most 0 with no condition on y. Second, the extremes: at y = π the left side is -sin x ≤ 0 and the right side is sin x ≥ 0, the widest margin anywhere; for small x and y the difference is about -(xy(x+y))/2, negative but tiny, so the tightest case is near the corner x = y = 0 where equality is approached. Both ends behave, and the equality set is exactly x = 0, y = 0, or x = y = π. It is worth flagging that the common one-line claim "equality only when cos x = cos y = 1" is not quite right: x = 0 with any y, and the pair x = y = π, also give equality. That does not change the answer, since equality still satisfies ≤ .
💡Key takeaway

Expand sin(x+y) and move everything to one side: the difference is sin x(cos y - 1) + sin y(cos x - 1). On [0, π] sine is never negative and cosine never beats 1, so both products are nonpositive and the inequality can never fail — the answer is the whole interval [0, π], choice (E).

  • Read the choices as one threshold
  • Ask how it could fail instead
  • Expand the compound angle
  • Move everything to one side and regroup
  • Pin down the sign of each factor
  • Both products are nonpositive
  • Check the boundary and the equality case
  • Name the largest subset

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