One root of the equation $\cos x - x + \frac{1}{2} = 0$ lies in the interval
$\left[ {0,\,\frac{\pi }{2}} \right]$
$\left[ { - \frac{\pi }{2},\,0} \right]$
$\left[ {\frac{\pi }{2},\,\pi } \right]$
$\left[ {\pi ,\frac{{3\pi }}{2}} \right]$
The number of $x \in [0,2\pi ]$ for which $\left| {\sqrt {2\,{{\sin }^4}\,x\, + \,18\,{{\cos }^2}\,x} - \,\sqrt {2\,{{\cos }^4}\,x\, + \,18\,{{\sin }^2}\,x} } \right| = 1$ is
Let $f:[0,2] \rightarrow R$ be the function defined by
$f ( x )=(3-\sin (2 \pi x )) \sin \left(\pi x -\frac{\pi}{4}\right)-\sin \left(3 \pi x +\frac{\pi}{4}\right)$
If $\alpha, \beta \in[0,2]$ are such that $\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]$, then the value of $\beta-\alpha$ is. . . . . . . . .
For $0<\theta<\frac{\pi}{2}$, the solution(s) of $\sum_{m=1}^6 \operatorname{cosec}\left(\theta+\frac{(m-1) \pi}{4}\right) \operatorname{cosec}\left(\theta+\frac{m \pi}{4}\right)=4 \sqrt{2}$ is(are)
$(A)$ $\frac{\pi}{4}$ $(B)$ $\frac{\pi}{6}$ $(C)$ $\frac{\pi}{12}$ $(D)$ $\frac{5 \pi}{12}$
If $\sin 2x + \sin 4x = 2\sin 3x,$ then $x =$
If $\sin \theta + \cos \theta = \sqrt 2 \cos \alpha $, then the general value of $\theta $ is