The number of solutions of the equation $\sqrt[3]{{\sin \theta  - 1}} + \sqrt[3]{{\sin \theta }} + \sqrt[3]{{\sin \theta  + 1}} = 0$ in $[0,4\pi]$ is 

The number of solutions of equation $3cos^2x - 8sinx = 0$ in $[0, 3\pi]$ is

If $\cot \theta + \cot \left( {\frac{\pi }{4} + \theta } \right) = 2$, then the general value of $\theta $ is

The number of solutions to $\sin x=\frac{6}{x}$ with $0 \leq x \leq 12 \pi$ is

If $S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\begin{array}{*{20}{c}}
0&{\cos {\mkern 1mu} x}&{ - \sin {\mkern 1mu} x}\\
{\sin {\mkern 1mu} x}&0&{\cos {\mkern 1mu} x}\\
{\cos {\mkern 1mu} x}&{\sin {\mkern 1mu} x}&0
\end{array}} \right| = 0} \right\},$ then $\sum\limits_{x \in S} {\tan \left( {\frac{\pi }{3} + x} \right)} $ is equal to

If $\tan (\pi \cos \theta ) = \cot (\pi \sin \theta )$, then $\sin \left( {\theta + \frac{\pi }{4}} \right)$ equals