Answer
$\{ 0.3876, 2.7540, 3.5292, 5.8956 \}$
Work Step by Step
Step 1. We can solve the equation for $sin(x)$ as $sin(x)=\pm\sqrt {\frac{1}{7}}\approx\pm0.3780$
Step 2. For $sin(x)=0.3780$, we can find the reference angle as $x_0=sin^{-1}(0.3780)\approx0.3876$
and all x-values in $[0,2\pi)$ as $x\approx0.3876$ and $x=\pi-x_0\approx2.7540$
Step 3. For $sin(x)=-0.3780$, we can find the reference angle as $x_0=sin^{-1}(0.3780)\approx0.3876$
and all x-values in $[0,2\pi)$ as $x=\pi+x_0\approx3.5292$ and $x=2\pi-x_0\approx5.8956$
Step 4. The solutions for the original equation in $[0,2\pi)$ are $\{ 0.3876, 2.7540, 3.5292, 5.8956 \}$
