The sides of a triangle inscribed in a given circle subtend angles $$\alpha $$, $$\beta $$ and $$\gamma $$ at the centre. The minimum value of the arithmetic mean of $$cos\left[ {\alpha + {\pi \over 2}} \right],\,\cos \left[ {\beta + {\pi \over 2}} \right]$$ and $$cos\left[ {\gamma + {\pi \over 2}} \right]$$ is equal to _______.

Find the area bounded by the curves, $${x^2} + {y^2} = 25,\,4y = \left| {4 - {x^2}} \right|$$ and $$x=0$$ above the $$x$$-axis.

The solution set of the system of equations $$X + Y = {{2\pi } \over 3},$$ $$cox\,x + cos\,y = {3 \over 2},$$ where x and y are real, is _____.

The set of all $$x$$ in the interval $$\left[ {0,\,\pi } \right]$$ for which $$2\,{\sin ^2}x - 3$$ $$\sin x + 1 \ge 0,$$ is _____.