2007 AMC 12A Problem 17 — Solution | LIVE by Po-Shen Loh

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2007 AMC 12A Problem 17

Below is the professionally curated solution for Problem 17 of the 2007 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 12A solutions, or check the answer key.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:trigonometric identity algebraic manipulation

Difficulty rating: 1570

17.

Suppose that $\sin a+\sin b=\sqrt{\tfrac53}$ and $\cos a+\cos b=1.$ What is $\cos(a-b)?$

$\sqrt{\dfrac53}-1$

$\dfrac13$

$\dfrac12$

$\dfrac23$

$1$

Solution:

Squaring both equations gives $\sin^2 a+2\sin a\sin b+\sin^2 b=\tfrac53$ and $\cos^2 a+2\cos a\cos b+\cos^2 b=1.$

Adding and using $\sin^2\theta+\cos^2\theta=1$ twice, $2+2(\sin a\sin b+\cos a\cos b)=\tfrac83.$

So $\cos(a-b)=\sin a\sin b+\cos a\cos b=\tfrac13.$

Thus, the correct answer is B.

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