sin(2x)=2sin(x)cos(x). If tan(x) is 12/5 then we can see a right-angled triangle where the hypotenuse is sqrt(12^2+5^2)=sqrt(169)=13. Therefore, sin(x)=12/13 and cos(x)=5/13. Therefore sin(2x)=2*5*12/169=120/169.
The range for x is between quadrants 1 and 3 and tangent is positive in quadrants 1 and 3 only. If the angle is between 180 and 270 (quadrant 3) then doubling it gives us an angle between 360 and 540. If we subtract 360 from this range we get 0 and 180. This range is quadrants 1 and 2 where sine is positive, so sin(2x) will also be positive.