In $\triangle PQR$,let $\angle P > \angle Q$. If the radian measures of $\angle P$ and $\angle Q$ satisfy the equation $4 \sin^3 x - 3 \sin x + a = 0$ where $0 < a < 1$,then the radian measure of $\angle R$ is

$A$ tower $AB$ leans towards west making an angle $\alpha$ with the vertical. The angular elevation of $B$,the top most point of the tower,is $\beta$ as observed from a point $C$ due east of $A$ at a distance $d$ from $A$. If the angular elevation of $B$ from a point $D$ due east of $C$ at a distance $2d$ from $C$ is $\gamma$,then $2\tan \alpha$ can be given as

Statement $-1$: The number of common solutions of the trigonometric equations $2\sin^2\theta - \cos 2\theta = 0$ and $2\cos^2\theta - 3\sin\theta = 0$ in the interval $[0, 2\pi]$ is two.
Statement $-2$: The number of solutions of the equation $2\cos^2\theta - 3\sin\theta = 0$ in the interval $[0, \pi]$ is two.

In a $\Delta ABC$,$a = a_1 = 2$,$b = a_2$,$c = a_3$ such that $a_{p+1} = \frac{5^p}{3^{2-p}} a_p \left( 2^{2-p} - \frac{4p-2}{5^p} a_p \right)$ where $p = 1, 2$,then:

The set of values of $a$ for which the equation $\cos\, 2x + a\, \sin\, x = 2a - 7$ possesses a solution is:

In $\triangle ABC$,if $x=\tan \left(\frac{B-C}{2}\right) \tan \frac{A}{2}$,$y=\tan \left(\frac{C-A}{2}\right) \tan \frac{B}{2}$,and $z=\tan \left(\frac{A-B}{2}\right) \tan \frac{C}{2}$,then $(x+y+z)$ is equal to