2007 AMC 12A Problems/Problem 17

Problem

Suppose that $\sin a + \sin b = \sqrt{\frac{5}{3}}$ and $\cos a + \cos b = 1$ . What is $\cos (a - b)$ ?

$\mathrm{(A)}\ \sqrt{\frac{5}{3}} - 1\qquad \mathrm{(B)}\ \frac 13\qquad \mathrm{(C)}\ \frac 12\qquad \mathrm{(D)}\ \frac 23\qquad \mathrm{(E)}\ 1$

Solution

We can make use the of the Pythagorean identities: square both equations and add them up:

$\sin^2 a + \sin^2 b + 2\sin a \sin b + \cos^2 a + \cos^2 b + 2\cos a \cos b = \frac{5}{3} + 1$
$2 + 2\sin a \sin b + 2\cos a \cos b = \frac{8}{3}$
$2(\cos a \cos b + \sin a \sin b) = \frac{2}{3}$

This is just the cosine difference identity, which simplifies to $\cos (a - b) = \frac{1}{3} \Longrightarrow \mathrm{(B)}$

2007 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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2007 AMC 12A Problems/Problem 17

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