If $\sin \,\theta  + \sqrt 3 \cos \,\theta  = 6x - {x^2} - 11,x \in R$ , $0 \le \theta  \le 2\pi $ , then the equation has solution for

If $\sin \theta = \sqrt 3 \cos \theta , - \pi < \theta < 0$, then $\theta = $

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

The most general value of $\theta $ satisfying the equations $\tan \theta = - 1$ and $\cos \theta = \frac{1}{{\sqrt 2 }}$ is

Find the principal and general solutions of the equation $\sec x=2$

Number of solution $(s)$ of equation $cosec\, \theta -cot \,\theta = 1$ in $[0,2 \pi]$ is-