How many angles \theta with 0≤\theta\le2π satisfy \log(\sin(3\theta))+\log(\cos(2\theta))=0?

2024 AMC 12A Problem 8 — Solution

2024 AMC 12A Problem 8

Attempt Problem 8 of the 2024 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 12A solutions, or check the answer key.

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How many angles $\theta$ with $0\le\theta\le2\pi$ satisfy $\log(\sin(3\theta))+\log(\cos(2\theta))=0?$

$0$

$1$

$2$

$3$

$4$

Answer: A

Concepts:logarithm trigonometry bounding to limit cases

Difficulty rating: 1480

Solution:

The equation means $\sin(3\theta)\cos(2\theta)=1$ with both factors positive (for the logs to be defined). Since $\sin(3\theta)\le1$ and $\cos(2\theta)\le1,$ their product is $1$ only if $\sin(3\theta)=1$ and $\cos(2\theta)=1$ simultaneously. But $\cos(2\theta)=1$ forces $\theta\in\{0,\pi,2\pi\},$ where $\sin(3\theta)=0\ne1.$ No angle works. Thus, the correct answer is A.

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