Maximizing Area of a Trigonometric Rectangle

A rectangle has side lengths $ \sin x $ and $ \cos 2x $, where $ x $ is a real number which makes both $ \sin x $ and $ \cos 2x $ positive. What is the greatest possible area of the rectangle? The answer is in the form $\frac{a\sqrt{b}}{c}$ where $a$, $b$, $c$ are positive integers, $b$ is square-free, and $\gcd(a,c) = 1$. Input the value of $a+b+c$.