HMMT February

Algebra Difficulty 4.9 AIME Find the answer

Given that $x+\sin y=2008$ and $x+2008 \cos y=2007$ , where $0 \leq y \leq \pi / 2$ , find the value of $x+y$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Subtracting the two equations gives $\sin y-2008 \cos y=1$ . But since $0 \leq y \leq \pi / 2$ , the maximum of $\sin y$ is 1 and the minimum of $\cos y$ is 0 , so we must have $\sin y=1$ , so $y=\pi / 2$ and $x+y=2007+\frac{\pi}{2}$ .

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