let use # to represent theta
Given that sin(#) = -12/13
# is in Quadrant III. we are to find sin(2#)
since sin(#) = opposite/hypotenus
Then, using pythagoras theorem
13^2 = 12^2 - x^2
169 = 144 - x^2
collecting like terms
x^2 = 169 - 144
x^2 = 25
x = 5
Therefore, cos(#) = -5/13
sin(2#) = 2sin#cos# (trigonometric rule)
sin(2#) = 2x - (12/13)x - (5/13)
= 120/169