Question

Find the domain of definition of the following function:
$y=\sqrt{-2{\cos }^{2}x+3\cos x-1}$

Answer 1

$-2{\cos }^{2}x+3\cos x-1\ge 0$

$-[2{\cos }^{2}x-3\cos x+1]\ge 0$

$-[2{\cos }^{2}x-2\cos x-\cos x+1]\ge 0$

$-\left[2\cos x\right(\cos x-1)-1(\cos x-1\left)\right]\ge 0$

$-(2\cos x-1)(\cos x-1)\ge 0$

$-(\cos x-\frac{1}{2})(\cos x-1)\ge 0$

$(\cos x-\frac{1}{2})(\cos x-1)\le 0$

$\therefore$ $\cos x\in [\frac{1}{2},1].$

$x\in [2n\pi -\frac{\pi }{3},2n\pi +\frac{\pi }{3}].$