If sin x=3cos x, then what is sin xcos x?

1988 AMC 12 Problem 13

Attempt Problem 13 of the 1988 AMC 12 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 1988 AMC 12 solutions, or check the answer key.

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13.

\sin x=3\cos x,

then what is

\sin x\cos x?

\frac16

\frac15

\frac29

\frac14

\frac3{10}

Square the given relation and use

\sin^2x+\cos^2x=1

The given relation also determines the sign of the product

Squaring gives

\sin^2x=9\cos^2x.

Hence

10\cos^2x=1.

Multiplying

\sin x=3\cos x

\cos x

gives

\sin x\cos x=3\cos^2x=\frac{3}{10}.

Thus the correct answer is E.