AMC 10 · 2024 · #8
Grade 11 algebraPick an answer.
There is no way to peel θ out of log(sin(3θ)) + log(cos(2θ)) = 0 by ordinary algebra, so the plan is to squeeze the equation from both sides until only a short list of candidates is left. Tool #15 (Organize Information in More Ways) does the rewriting: a sum of two logs is the log of a product, and a log that equals 0 means its input is exactly 1, so the log equation becomes the single statement sin(3θ)cos(2θ) = 1. Tool #14 (Extreme Principle) does the real work. Sine and cosine can never exceed 1, and the domain of the logarithm forces both factors to be positive, so a product equal to 1 is only possible when each factor is sitting exactly at its own maximum. That is a boundary case, and boundary cases come with very few candidates. Tool #3 (Eliminate Possibilities) closes it out: solve the easier of the two maximum conditions, write down the handful of angles it permits, and test each one against the other condition.
Find where the logs are legal
Both quantities must be positive.
The equation can only talk about angles where both logs actually exist, so the domain is the first fact to write down, not the last.
9.F-IF.A.1Organize Information In More WaysTurn the log equation into a product
Merged, their product is 1.
Adding logs multiplies what is inside them, and a log worth 0 means the inside is exactly 1.
11.F-LE.A.4Organize Information In More WaysA product of 1 pins both factors
Both must be maximal, so each equals 1.
Two positive numbers that are each capped at 1 can only multiply to 1 if neither one gives anything away, so both must sit exactly at the cap.
Two positive numbers each capped at one can multiply to one only if neither gives anything away.
▸ Why?
If either factor slipped below one the product would slip below one too.
▸ Why?
A product held at its ceiling forces both factors to sit exactly at their own ceilings.
List every angle with cosine at its peak
Only three candidates: 0, π, 2π.
Cosine only hits its peak after a whole number of full turns, so doubling the angle fits just three full turns inside the interval.
11.F-TF.A.2Eliminate PossibilitiesTest the candidates and count
All three fail the sine test, leaving 0 angles.
Cosine peaks exactly at the angles where sine is passing through zero, so the two demands point at opposite spots on the unit circle.
11.F-TF.A.2Eliminate PossibilitiesIf two positive numbers that are each at most 1 multiply to 1, both have to be exactly 1 — and cos(2θ) = 1 only at multiples of π, which is precisely where sin(3θ) is stuck at 0.
- Find where the logs are legal
- Turn the log equation into a product
- A product of 1 pins both factors
- List every angle with cosine at its peak
- Test the candidates and count