Answer
$\{-1.895,0,1.895\}$
Work Step by Step
We are given the equation:
$x=2\sin x$
One obvious solution is
$x=0$
Because $-1\leq \sin x\leq 1$, we have:
$-2\leq\sin x\leq 2$
$-2\leq x\leq 2$
So, if there are other solutions, they are in the interval $[-2,2]$.
The function $y=x$ is increasing on $[0,2]$, having $y=2$ for $x=2$. The function $y=2\sin x$ is increasing on $\left[0,\dfrac{\pi}{2}\right]$, when $2\sin\dfrac{\pi}{2}=2$, and decreasing on $\left[\dfrac{\pi}{2},2\right]$. Therefore, the two graphs must intersect once in the interval $\left[\dfrac{\pi}{2},2\right]$.
Because of symmetry, the same goes for the interval $[-2,0]$. So the equation has 3 solutions. We use a graphing utility to approximate them:
$\{-1.895,0,1.895\}$
