The number of solutions of the equation $2x + 3 \tan x = \pi$,where $x \in [-2\pi, 2\pi] - \left\{ \pm \frac{\pi}{2}, \pm \frac{3\pi}{2} \right\}$,is

The sum of all values of $x$ in $[0, 2\pi]$,for which $\sin x + \sin 2x + \sin 3x + \sin 4x = 0$,is equal to: (in $\pi$)

If $\theta \in \left[-\frac{7 \pi}{6}, \frac{4 \pi}{3}\right]$,then the number of solutions of $\sqrt{3} \operatorname{cosec}^2 \theta - 2(\sqrt{3}-1) \operatorname{cosec} \theta - 4 = 0$ is equal to

The number of roots of the equation $\cos^7 \theta - \sin^4 \theta = 1$ that lie in the interval $[0, 2\pi]$ is

The solution of $\frac{1}{2} + \cos x + \cos 2x + \cos 3x + \cos 4x = 0$ is:

If $2\sin \theta + \tan \theta = 0$,then the general values of $\theta$ are