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 \(M+A+B\).