AMC 10 · 2021 · #19
Grade 11 algebraPick an answer.
You cannot compare a sine to a cosine directly, so Tool #15 (Organize Information in More Ways) does the load-bearing move: rewrite the cosine side as a sine using the cofunction identity, so both sides speak the same language. Normally "sin A = sin B" opens up infinitely many branches (A = B + 2π k and A = π - B + 2π k), which is where this problem eats most students. Tool #14 (Extreme Principle) closes that door first: check the extreme values of sin x and cos x on [0, π] and you find both inner angles are trapped inside [-π/2, π/2], the window where sine is one-to-one. Then the angles must simply be equal, Tool #13 (Convert to Algebra) reduces everything to sin x + cos x = 1, and Tool #2 (Make a Systematic List) reads off the finitely many solutions in the interval. Tool #3 (Eliminate Possibilities) is what kills the unwanted branches once the angles are trapped, and Tool #6 (Guess and Check) closes the job by substituting the two finalists back into the original equation.
Trap both inner angles
Both inner angles sit in a narrow range.
Pinning down exactly which window each angle lives in is what will later let us cancel the sine without spawning extra branches.
11.F-TF.A.2Extreme PrincipleRewrite the cosine as a sine
Put both sides in the same function.
Sine and cosine are the same wave shifted by a quarter turn, so either one can always be dressed up as the other.
Sine and cosine are the same wave shifted by a quarter turn, so either can be dressed as the other.
▸ Why?
Adding a fixed angle to another is what a shifted wave records, so the shift is just an angle sum.
▸ Why?
A quarter turn is a fixed share of the full circle, so the shift is exactly the same everywhere.
Cancel the sine safely
The range lets us cancel it safely.
Equal sines only force equal angles when both angles are locked in the stretch where the sine wave never repeats a value.
11.F-BF.B.4Eliminate PossibilitiesReduce to a clean equation
It reduces to sine plus cosine equals one.
A messy nested trig equation has collapsed into one of the friendliest equations in trigonometry.
9.A-REI.B.3Convert To AlgebraFold into one wave
Fold the sum into one sine wave.
Adding a sine and a cosine of the same angle always produces one shifted sine wave, and one wave is far easier to count against.
11.F-TF.C.9Organize Information In More WaysList the solutions in range
List the solutions in range.
A horizontal line cuts one hump of a sine wave in at most two places, and this window covers exactly one hump plus a tail.
11.F-TF.A.2Make A Systematic ListVerify and count
Verifying gives 2.
Plugging the finalists back in costs seconds and catches any solution that only survived because of an algebraic shortcut.
11.A-REI.D.11Guess And CheckWhen a sine is set equal to a cosine, first rewrite one as the other so both sides are the same function; then check whether both angles are trapped in [-π/2, π/2], because there the sine never repeats a value and the angles themselves must be equal. That collapses this monster into sin x + cos x = 1, which holds exactly at x = 0 and x = π/2 on [0, π] — so the answer is (C).
- Trap both inner angles
- Rewrite the cosine as a sine
- Cancel the sine safely
- Reduce to a clean equation
- Fold the sum into one wave
- List the solutions in range
- Verify and count