Final answer:
To find tan(θ/2), use the given information about sin(θ) and cos(θ) in Quadrant III. Substitute the values into the formula for tan(θ/2) = sin(θ)/(1 + cos(θ)) to get the result.
Step-by-step explanation:
To find tan(θ/2), we need to use the given information that sin(θ) = -12/13 and θ is in Quadrant III. We know that tan(θ) = sin(θ)/cos(θ). Since sin(θ) is negative in Quadrant III, we can use the identity sin(θ) = -sqrt(1 - cos²(θ)) to find the value of cos(θ). Therefore, cos(θ) = -sqrt(1 - (-12/13)²) = -sqrt(1 - 144/169) = -sqrt(25/169) = -5/13.
Now, we can substitute the values of sin(θ) and cos(θ) into the formula for tan(θ/2) = sin(θ)/(1 + cos(θ)) to find tan(θ/2).
tan(θ/2) = (-12/13)/(1 + (-5/13)) = (-12/13)/(13/13 - 5/13) = (-12/13)/(8/13) = -12/8 = -3/2.