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Revision as of 20:31, 3 July 2013

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 $\cos (a - b) = \frac{1}{3} \Longrightarrow \mathrm{(B)}$

See also

2007 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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