The correct answer is $\boxed{\text{B}}$.
The equation $\sin(x) = \frac{x}{2}$ has two distinct solutions in the interval $[0, 2\pi]$. These solutions are $x = \frac{\pi}{6}$ and $x = \frac{5\pi}{6}$.
To see this, we can use the graph of the sine function. The graph of the sine function is shown below.
[asy]
unitsize(1 cm);
draw((0,-1.2)–(0,1.2));
draw((0,0)–(2*pi,0));
real ticklen=0.1;
real tickspace=0.2;
real axisarrowsize=0.14inch;
real ticklength=0.1cm;
real vectorarrowsize=0.2cm;
draw((0,-1.2)–(0,1.2));
draw((0,0)–(2*pi,0));
label(“$x$”, (2*pi,0), S);
label(“$y$”, (0,1.2), E);
real i;
for (i=-pi; i<pi; i+=pi/4) {
draw((i,-0.1)–(i,0.1));
}
real j;
for (j=-1.2; j<1.2; j+=0.2) {
if (j>-0.1 && j<0.1) continue;
draw((0,j)–(0.1,j));
}
dot(“$\frac{\pi}{6}$”, (pi/6,0.1), S);
dot(“$\frac{5\pi}{6}$”, (5*pi/6,0.1), S);
[/asy]
The graph intersects the line $y = \frac{x}{2}$ at $x = \frac{\pi}{6}$ and $x = \frac{5\pi}{6}$. Therefore, these are the two distinct solutions to the equation $\sin(x) = \frac{x}{2}$ in the interval $[0, 2\pi]$.