First define the function f(x)=sinx - 2x+3, this is continuous and differentiable function for all x.
Now, f(0)=sin0-2*0+3>0 and f(6)=sin(6)-12+3 ~ 1-12+3<0, from the Intermediate value theorem follows that there exist x_0 in (0,6) so that f(x_0)=0, hence we know that f(x) have one zero.
Now we will prove that above zero is unique!
We look at the derivative of f(x).
f'(x)=cosx-2<0 for all x. In other words the function f(x) is decreasing for every x, therefor the function is one-to-one, hence f(x)=0 only at x_0.