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Shireen's Hard Trig Problem

From http://mathteachermambo.blogspot.com/2011/12/challenge.html Shireen Dadmehr writes: "This problem makes me happy in its difficulty and challenge. I threw down the gauntlet to my precalculus students. We'll see if they run with it to the finish line or not. Either way, I wanted the smartie smarts to at least know there are some problems out there that require more than just rote brain power."