How many angles θ with 0≤θ≤2π satisfy log(sin(3θ))+log(cos(2θ))=0?

2024 AMC 12A Problem 8

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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

\log(\sin3\theta)+\log(\cos2\theta)

=\log(\sin3\theta\cos2\theta),

which must equal

0

A product of two numbers each at most

1

equals

1

only when both equal

1

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.