If $\sin \theta + \cos \theta = 1$,then the general value of $\theta$ is

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 most general value of $\theta$ which satisfies both the equations $\sin \theta = -\frac{1}{2}$ and $\tan \theta = \frac{1}{\sqrt{3}}$ is:

The number of solutions of the equation $4 \cos 2 \theta \cos 3 \theta = \sec \theta$ in the interval $[0, 2 \pi]$ is

The smallest positive angle which satisfies the equation $2\sin^2 \theta + \sqrt{3} \cos \theta + 1 = 0$ is

Let $S$ be the sum of all solutions (in radians) of the equation $\sin^{4} \theta + \cos^{4} \theta - \sin \theta \cos \theta = 0$ in $[0, 4\pi]$. Then $\frac{8S}{\pi}$ is equal to ...... .