If $0 \leq x \leq 2 \pi$,then the number of real values of $x$ which satisfy the equation $\sin x + \sin 2x + \sin 3x + \sin 4x = 0$ is

The sum of solutions in $x \in (0, 2\pi)$ of the equation $4\cos(x)\cos\left(\frac{\pi}{3} - x\right)\cos\left(\frac{\pi}{3} + x\right) = 1$ is equal to: (in $\pi$)

If angle $\theta$ in $[0, 2\pi]$ satisfies both the equations $\cot \theta = \sqrt{3}$ and $\sqrt{3} \sec \theta + 2 = 0$,then $\theta$ is equal to

Let $\theta \in [0, 4\pi ]$ satisfy the equation $(\sin \theta + 2) (\sin \theta + 3) (\sin \theta + 4) = 6$. If the sum of all the values of $\theta$ is of the form $k\pi$,then the value of $k$ is:

If $\sqrt{3} \cos \theta + \sin \theta = \sqrt{2}$,then the most general value of $\theta$ is

The equation $3 \cos x + 4 \sin x = 6$ has