SineCosine at its pinnacle

Geometry Level 2

Let the maximum value of $\sin x \cos x$ for $x \in \mathbb{R}$ be $M$ . If the value of $x$ for which $\sin x \cos x$ is maximum, where $0< x < \pi$ , can be expressed as $\dfrac{A\pi}{B}$ for positive coprime integers $A$ and $B$ , compute $M+A+B$ .