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 $\sin \theta = \sqrt{3} \cos \theta$ and $-\pi < \theta < 0$,then $\theta = $

The number of solutions of the equation $\text{sgn}(\sin x) = \sin^2 x + 2\sin x + \text{sgn}(\sin^2 x)$ in the interval $\left[ -\frac{5\pi}{2}, \frac{7\pi}{2} \right]$ is (where $\text{sgn}(\cdot)$ denotes the signum function):

The general value of $\theta$ satisfying the equation $\tan^2 \theta + \sec 2\theta = 1$ is

If $\cos 6\theta + \cos 4\theta + \cos 2\theta + 1 = 0$,where $0 < \theta < 180^\circ$,then $\theta = $

Let $S = \left[-\pi, \frac{\pi}{2}\right) - \left\{-\frac{\pi}{2}, -\frac{\pi}{4}, -\frac{3\pi}{4}, \frac{\pi}{4}\right\}$. Then the number of elements in the set $\{\theta \in S : \tan \theta(1 + \sqrt{5} \tan(2\theta)) = \sqrt{5} - \tan(2\theta)\}$ is: