Easiest way in general to find the sin, cos, arcsin, arccos, of "not so easy" angles/values without using a calculator? I was wondering if there are any easy ways in general to find the sin, cos, arcsin, arccos, of "not so easy" angles/values without using a calculator. By "not so easy" I mean just not things like 0, $\pi/6$, $\pi/4$, $\pi/3$, or $\pi/2$, which one encounters routinely. And also not $\sin(\theta+2n\pi)$, $\sin(n\pi)$, $\sin(-\theta)$, and the like. The reason I'm asking this terribly general question (please forgive me for that) is because in Calculus class I always make some sort of error when these sorts of calculations need to be done to solve a problem.
For example, on a recent question I had to find $0<\theta<2\pi$ satisfying $\sin\theta=-1/2$. 
We know $\sin(\pi/6)=1/2$ and $\sin(-\pi/6)=-1/2$ since $-\sin(x)=\sin(-x)$. Also we know sin has period $2\pi$ so $\sin(-\pi/6+12\pi/6)=\sin(11\pi/6)$=$-1/2$. 
But how would one get $\sin(7\pi/6)=-1/2$ for example?
Or similarly how would one know the value of sin $4\pi/3$, $5\pi/3$, $5\pi/4$, $6\pi/5$, $5\pi/6$, etc. Or cos, arcsin, arccos, of such "nasty" values?
Again, I apologize for the ridiculously general question, it's just that I know no other way of asking it.

Since $\pi/4$ is an angle of a $1:1:\sqrt{2}$ right-angled triangle, and $\pi/6$ and $\pi/3$ are angles of a $1:\sqrt{3}:2$ right-angled triangle, the $\sin / \cos / \tan$ of these and their multiples are easy to calculate on the fly.
$\sin 6\pi/5$  or something similar would be harder.
