Trigonometric Equations

If $6\left|\sin x\right|=x$ when $x\in \left[0,2\pi \right]$, then the number of solutions are 

The number of solutions of equation $\left|\tan 2x\right|=\sin x$ in $\left[0,\pi \right]$ is

The number of distinct real roots of the equation $x={\left(\frac{5\pi }{2}\right)}^{\cos x}$is:

If $\frac{\sin 3\theta }{\cos 2\theta }<0$, then $\theta$lies in:

If $2{\sin }^2\left(x-\frac{\mathrm{\pi }}{3}\right)-5\sin \left(x-\frac{\mathrm{\pi }}{3}\right)+2<0$ then $x$ belongs to

If the equation $x^2+2x+2+e^{\alpha }-2\sin \beta =0$ has a real solution, then $(n\in Z)$:

The function $f\left(x\right)=\sqrt{3}\sin x-\cos x$ will increase monotonically in the interval(s):

If $\cos 3\theta +\sin 3\theta +(2\sin 2\theta -3)(\sin \theta -\cos \theta )>0,$ then $\theta$ lies in