AMC12 2006 12A, Problem 15

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

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} $. $\fbox{\frac{\pi}{6}}$.

Answer: \frac{\pi}{6}