To solve this problem, let’s go through the steps carefully.

Problem Analysis

We are given:

• The terminal side of the angle $\theta$ in the third quadrant (since $180^\circ < \theta < 270^\circ$), with the coordinates $(-4, -3)$ as a point on the terminal side.
• We are asked to find the value of $\sin 2\theta$.

Steps to Solution

1. Find $r$ (the radius or hypotenuse): $r = \sqrt{x^2 + y^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

2. Determine $\sin \theta$ and $\cos \theta$:

   • Since $\sin \theta = \frac{y}{r}$ and $\cos \theta = \frac{x}{r}$, $\sin \theta = \frac{-3}{5} \quad \text{and} \quad \cos \theta = \frac{-4}{5}$

3. Use the Double-Angle Formula for Sine: $\sin 2\theta = 2 \sin \theta \cos \theta$ Substituting the values we found: $\sin 2\theta = 2 \cdot \frac{-3}{5} \cdot \frac{-4}{5} = 2 \cdot \frac{12}{25} = \frac{24}{25}$

Answer

The value of $\sin 2\theta$ is $\frac{24}{25}$, which matches option B.

Would you like further clarification on any of these steps or additional details?

Here are some related questions you might explore:

1. How would we find $\cos 2\theta$ given the same information?
2. What is the value of $\tan 2\theta$ for the same angle?
3. Can we generalize this approach for any point in the third quadrant?
4. How does the double-angle identity for sine differ from other double-angle identities?
5. Why do the values of $\sin \theta$ and $\cos \theta$ have different signs in different quadrants?

Tip: Double-angle identities are useful in trigonometry when you know $\sin \theta$ or $\cos \theta$ but need to find expressions involving $2\theta$.