The positive integer value of $$n\, > \,3$$ satisfying the equation $${1 \over {\sin \left( {{\pi \over n}} \right)}} = {1 \over {\sin \left( {{{2\pi } \over n}} \right)}} + {1 \over {\sin \left( {{{3\pi } \over n}} \right)}}$$ is

The number of values of $$\theta $$ in the interval, $$\left( { - {\pi \over 2},\,{\pi \over 2}} \right)$$ such that$$\,\theta \ne {{n\pi } \over 5}$$ for $$n = 0,\, \pm 1,\, \pm 2$$ and $$\tan \,\theta = \cot \,5\theta \,$$ as well as $$\sin \,2\theta = \cos \,4 \theta $$ is

The maximum value of the expression $${1 \over {{{\sin }^2}\theta + 3\sin \theta \cos \theta + 5{{\cos }^2}\theta }}$$ is

The number of all possible values of $$\theta $$ where $$0 < \theta < \pi ,$$ for which the system of equations $$$\left( {y + z} \right)\cos {\mkern 1mu} 3\theta = \left( {xyz} \right){\mkern 1mu} \sin 3\theta $$$ $$$x\sin 3\theta = {{2\cos 3\theta } \over y} + {{2\sin 3\theta } \over z}$$$ $$$\left( {xyz} \right){\mkern 1mu} \sin 3\theta = \left( {y + 2z} \right){\mkern 1mu} \cos 3\theta + y{\mkern 1mu} sin3\theta $$$

have a solution $$\left( {{x_0},{y_0},{z_0}} \right)$$ with $${y_0}{z_0}{\mkern 1mu} \ne {\mkern 1mu} 0,$$ is