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2006 AMC 12A Problems/Problem 15

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Problem

Suppose \cos x=0 and \cos (x+z)=1/2. What is the smallest possible positive value of z?

\mathrm{(A) \ } \frac{\pi}{6}\qquad \mathrm{(B) \ } \frac{\pi}{3}\qquad \mathrm{(C) \ } \frac{\pi}{2}\qquad \mathrm{(D) \ } \...

Solution

  • For \cos x = 0, x must be in the form of \frac{\pi}{2} + \pi n, where n denotes any integer.
  • For \cos (x+z) = 1 / 2, x + z = \frac{\pi}{3} +2\pi n, \frac{5\pi}{3} + 2\pi n.

The smallest possible value of z will be that of \frac{5\pi}{3} - \frac{3\pi}{2} = \frac{\pi}{6} \Rightarrow A.

See also

2006 AMC 12A (ProblemsResources)
Preceded by
Problem 14
Followed by
Problem 16
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
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