Method of Characteristics and Initial Value Problem $u_t + 3u_x = 2t$, $u(x,0)=\sin(x/2)$. 

I used the method of characteristics to get the answer, $u(x,t)=t^2 + 2\sin^{-1}(x-3t)$. 

Does this satisfy the initial condition? I checked for the first equation and it does; however I do not think it satisfies the intial value when $t=0.$ Am I correct in saying so?
The solution is $u=\frac{2}{3}\cos(\frac{x}{2})+\sin(\frac{x-3t}{2})-\frac{2}{3}\cos(\frac{x-3t}{2})$.
answer for above question is $$u(x,t)=t^2+\sin(x−3t/2)$$