First note that 13π/12 = π + π/12 which puts it in quadrant III. That tells you that sin(13π/12) = –sin(π/12). So all you need to do now is find that.
Second, notice that π/3 – π/4 = π/12. So to find sin (π/12) find instead sin (π/3 –π.4). There's a formula for that. Use the sin(A – B) forrmula. Because you know the exact values of sine and cosine of those two angles, all that remains is a little arithmetic/algebra.
I have told you all the details you need to proceed. What I left out are things everyone studying trig needs to know