Sin-o-metry

There should definitely be an easier answer but my approach was analyzing all the possible solutions.As hard as it may seem, it took me only 3-4 mins to find them all out using the facts below:

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1. First off , note that $sinx = 0$ has only one answer in the given interval and that is $x=\pi$ ,keeping that in mind ,for the next equations we don't have to count the case of $x = \pi$ as in $sin2x=0 \rightarrow x=\frac {2\pi} {2} = \pi$ .

2. Start solving the equations one by one from 1 to 12 not in any other order.How does that help?Well you can see that if - for example - you are analyzing the case of $sin12x = 0$ you just need to write down all the fractions from $\frac{11}{12} \dots \frac{1}{12}$ but whenever a fraction is reducible be quite sure that , that case has already been counted.For example not long after starting your analysis you'll be analyzing $\frac {10}{12}$ which can be reduced to $\frac{5}{6}$ and this case is already analyzed in $sin6x=0$

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I don't know if I was able to convey my meaning to you well enough or not but if there was any problem understanding this just comment below my solution.I'll answer whenever possible.