AMC 10 · 2007 · #17
Grade 11 algebraPick an answer.
Solving for a and b looks hopeless, and it is also unnecessary. Tool #13 (Convert to Algebra) opens with the subtraction formula, which shows the target is built from cos acos b and sin asin b — two products. The load-bearing move is Tool #15 (Organize Information in More Ways): stop reading the two givens as two separate facts and combine them, because squaring a sum manufactures exactly the cross-product hiding inside it, and adding the two squared equations lets the Pythagorean identity swallow everything that is not wanted. That settles what the value must be. It does not settle that anything has that value, because squaring only runs one way, so Tool #7 (Identify Subproblems) splits the job in two: the forced value, and the existence of a pair. Tool #4 (Introduce a Variable) sets up half-angle variables s=(a+b)/2 and d=(a-b)/2 in which the system untangles, and Tool #11 (Work Backwards) uses them to build an explicit pair and plug it back in. Tool #3 (Eliminate Possibilities) is the ten-second filter that kills choice (E) before any of this starts.
See what the target is made of
The target expands into exactly the cross terms.
Knowing what the answer is built out of tells you exactly what to hunt for in the givens.
11.F-TF.C.9Convert To AlgebraSquare each given sum
Squaring each given sum produces those cross terms plus squares.
Squaring a sum conjures the product of its two terms out of nothing, which is the only way a product can come from an equation that only mentions a sum.
9.A-SSE.A.2Organize Information In More WaysAdd, and let the identity eat the squares
Adding lets the identity eat every square.
Pairing each sine square with the cosine square of the same angle is what makes every unwanted term vanish at once.
Pairing each sine square with the cosine square of the same angle makes every unwanted term vanish at once.
▸ Why?
Sine and cosine are the two legs of a right triangle with hypotenuse one, so their squares add to one.
▸ Why?
Squaring a sum spreads the multiplication over both terms, which is what conjures the product out of the sum.
Read off the forced value
Reading off gives 1/3.
An "if-then" only pays out when something actually satisfies the "if".
9.A-SSE.A.2Convert To AlgebraUntangle the system with half-angles
Half-angle forms untangle the original system.
Trading (a,b) for their half-sum and half-difference splits the system into a size part and a direction part.
11.F-TF.C.9Introduce A VariableBuild a pair that works
That builds an actual pair of angles.
A pair of numbers is a genuine (cosine, sine) of some real angle exactly when their squares add to 1.
11.F-TF.A.2Work BackwardsPlug it back in and finish
Substituting back confirms 1/3, choice (B).
A candidate that survives being substituted back turns "must be" into "is".
11.F-TF.C.9Identify SubproblemsSquare both given sums and add them: the Pythagorean identity swallows all four squares into a plain 2, and what is left over is exactly 2cos(a-b).
- See what the target is made of
- Square each given sum
- Add, and let the identity eat the squares
- Read off the forced value
- Untangle the system with half-angles
- Build a pair that works
- Plug it back in and finish