1. **State the problem:** We want to find an approximate numerical value for the expression $$\frac{\theta \tan 2\theta}{1 - \cos 3\theta}$$ using small angle approximations, where $\theta$ is small and measured in radians. 2. **Recall small angle approximations:** For small $\theta$, we use: - $\sin \theta \approx \theta$ - $\tan \theta \approx \theta$ - $\cos \theta \approx 1 - \frac{\theta^2}{2}$ 3. **Apply approximations to each part:** - $\tan 2\theta \approx 2\theta$ - $\cos 3\theta \approx 1 - \frac{(3\theta)^2}{2} = 1 - \frac{9\theta^2}{2}$ 4. **Rewrite the expression using approximations:** $$\frac{\theta \tan 2\theta}{1 - \cos 3\theta} \approx \frac{\theta \cdot 2\theta}{1 - \left(1 - \frac{9\theta^2}{2}\right)} = \frac{2\theta^2}{1 - 1 + \frac{9\theta^2}{2}} = \frac{2\theta^2}{\frac{9\theta^2}{2}}$$ 5. **Simplify the fraction:** $$\frac{2\theta^2}{\frac{9\theta^2}{2}} = 2\theta^2 \cdot \frac{2}{9\theta^2} = \frac{4\cancel{\theta^2}}{9\cancel{\theta^2}} = \frac{4}{9}$$ 6. **Final answer:** The approximate numerical value of the expression for small $\theta$ is $$\boxed{\frac{4}{9}}$$ This means the expression approaches $\frac{4}{9}$ as $\theta$ approaches zero.