Explanation

(d)
$\because\sin{x + 2\sqrt{2}\cos{x \geq \left( \sqrt{3} \right)^{2}}}$
$\Rightarrow \sin{x + 2\sqrt{2}\cos{x \geq 3}}$
$\Rightarrow \sin{\left( a + \cos^{- 1}\frac{1}{3} \right) \geq 1}$
$\Rightarrow \sin{\left( x + \cos^{- 1}\frac{1}{3} \right) = 1}$ $(\because\sin x\$cannot be greater than 1)
$\therefore x = n\pi + ( - 1)^{n}\frac{\pi}{2} - \cos^{- 1}\frac{1}{3}$
For solution in the interval $\lbrack - 2\pi,2\pi\rbrack,\ n = 0,\ 1,\ - 1,\ - 2$