The sides of a triangle are $\sin \alpha ,\,\cos \alpha $ and $\sqrt {1 + \sin \alpha \cos \alpha } $ for some $0 < \alpha < \frac{\pi }{2}$. Then the greatest angle of the triangle is.....$^o$

If both roots of quadratic equation ${x^2} + \left( {\sin \,\theta  + \cos \,\theta } \right)x + \frac{3}{8} = 0$ are positive and distinct then complete set of values of $\theta $ in $\left[ {0,2\pi } \right]$ is 

Let $A = \left\{ {\theta \,:\,\sin \,\left( \theta  \right) = \tan \,\left( \theta  \right)} \right\}$ and $B = \left\{ {\theta \,:\,\cos \,\left( \theta  \right) = 1} \right\}$ be two sets. Then

The general solution of $\sin x - \cos x = \sqrt 2 $, for any integer $n$ is

The number of elements in the set $S =\left\{\theta \in[0,2 \pi]: 3 \cos ^4 \theta-5 \cos ^2 \theta-2 \sin ^2 \theta+2=0\right\}$ is $...........$.

The number of values of $x$ in the interval $[0, 5 \pi  ] $ satisfying the equation $3{\sin ^2}x - 7\sin x + 2 = 0$ is