The number of solutions of the equation $\sqrt[3]{{\sin \theta - 1}} + \sqrt[3]{{\sin \theta }} + \sqrt[3]{{\sin \theta + 1}} = 0$ in $[0,4\pi]$ is
$2$
$4$
$5$
$6$
One of the solutions of the equation $8 \sin ^3 \theta-7 \sin \theta+\sqrt{3} \cos \theta=0$ lies in the interval
Let $S=\{\theta \in[0,2 \pi): \tan (\pi \cos \theta)+\tan (\pi \sin \theta)=0\}$.
Then $\sum_{\theta \in S } \sin ^2\left(\theta+\frac{\pi}{4}\right)$ is equal to
Let $P = \left\{ {\theta :\sin \,\theta - \cos \,\theta = \sqrt 2 \,\cos \,\theta } \right\}$ and $Q = \left\{ {\theta :\sin \,\theta + \cos \,\theta = \sqrt {2\,} \sin \,\theta } \right\}$ be two sets. Then
$\sum\limits_{r = 1}^{100} {\frac{{\tan \,{2^{r - 1}}}}{{\cos \,{2^r}}}} $ is equal to
$\sin 6\theta + \sin 4\theta + \sin 2\theta = 0,$ then $\theta = $