We are given that $\tan \theta = \frac{4}{3}$ and that $180^\circ < \theta < 270^\circ$. This range corresponds to the third quadrant, where both sine and cosine are negative, and tangent is positive.

Step-by-Step Solution:

1. Identify the Reference Triangle:

   • The tangent of an angle in a right triangle is given by $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$. Here, the opposite side is 4, and the adjacent side is 3. Since $\theta$ is in the third quadrant, both the opposite and adjacent sides will have negative values: $-4$ and $-3$.

2. Calculate the Hypotenuse: Use the Pythagorean theorem: $r = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

3. **Determine $\sin \theta$ and \