You don't have to memorize all of the stuff on the wikipedia page to figure out this sort of thing. For this question, you should know that $\sin(\pi/6)=1/2$ and either remember that $\sin(\pi + \theta)=-\sin(\theta)$ for any angle $\theta$, or be able to figure this out by remembering the graph of $\sin(\theta)$ and staring at the graph for a second or two. (I favour the latter approach). Or you can draw a unit circle and figure things out that way.
Anyway, taking $\theta=\pi/6$ in the formula $\sin(\pi+\theta)=-\sin(\theta)$, we get $\sin(7\pi/6)=\sin(\pi + \pi/6)=-\sin(\pi/6)=-1/2$.
If you think it's likely you'll be asked this kind of question, I would advise memorizing the values of $\sin$ and $\cos$ at $\pi/6$ and $\pi/3$ along with the graphs of $\sin$ and $\cos$, and practise using this to figure out other values of trig functions at various other angles.