Answer
$x=\frac{\pi}{8}$
$x=\frac{3\pi}{8}$
$x=\frac{9\pi}{8}$
$x=\frac{11\pi}{8}$
Work Step by Step
$2~sin~2x-\sqrt 2=0$
$2~sin~2x=\sqrt 2$
$sin~2x=\frac{\sqrt 2}{2}$
$2x=\frac{\pi}{4}$ or $2x=\frac{3\pi}{4}$
(with $0\lt2x\lt2\pi$)
If $x$ is in the interval $[0,2\pi)$, then $2x$ is in the interval $[0,4\pi)$
The period of $sin~θ$ is $2\pi$. So, we can add $2\pi$ to each answer above:
$2x=\frac{\pi}{4}+2\pi=\frac{9\pi}{4}$
$2x=\frac{3\pi}{4}+2\pi=\frac{11\pi}{4}$
Finally:
$2x=\frac{\pi}{4}$
$x=\frac{\pi}{8}$
$2x=\frac{3\pi}{4}$
$x=\frac{3\pi}{8}$
$2x=\frac{9\pi}{4}$
$x=\frac{9\pi}{8}$
$2x=\frac{11\pi}{4}$
$x=\frac{11\pi}{8}$
