The general value $\theta $ is obtained from the equation $\cos 2\theta = \sin \alpha ,$ is
$2\theta = \frac{\pi }{2} - \alpha $
$\theta = 2n\pi \pm \left( {\frac{\pi }{2} - \alpha } \right)$
$\theta = \frac{{n\pi + {{( - 1)}^n}\alpha }}{2}$
$\theta = n\pi \pm \left( {\frac{\pi }{4} - \frac{\alpha }{2}} \right)$
If the equation $2tan\ x \ sin\ x -2 tan\ x + cos\ x = 0$ has $k$ solutions in $[0,k \pi]$, then number of integral values of $k$ is-
If $\sin \theta + 2\sin \phi + 3\sin \psi = 0$ and $\cos \theta + 2\cos \phi + 3\cos \psi = 0$ , then the value of $\cos 3\theta + 8\cos 3\phi + 27\cos 3\psi = $
Let $A=\left\{\theta \in R:\left(\frac{1}{3} \sin \theta+\frac{2}{3} \cos \theta\right)^2=\frac{1}{3} \sin ^2 \theta+\frac{2}{3} \cos ^2 \theta\right\}$.Then
The number of solutions of the equation $2 \theta-\cos ^{2} \theta+\sqrt{2}=0$ is $R$ is equal to
The number of integral values of $k$, for which the equation $7\cos x + 5\sin x = 2k + 1$ has a solution, is