1. **State the problem:** Given $\sin \theta = -\frac{3}{5}$ and $\cos \theta = \frac{4}{5}$, find $\tan \theta$ for $0^\circ \leq \theta \leq 360^\circ$ without using a calculator. 2. **Recall the formula:** $\tan \theta = \frac{\sin \theta}{\cos \theta}$. 3. **Substitute the given values:** $$\tan \theta = \frac{-\frac{3}{5}}{\frac{4}{5}}$$ 4. **Simplify the fraction:** $$\tan \theta = -\frac{3}{5} \times \frac{5}{4}$$ 5. **Cancel common factors:** $$\tan \theta = -\frac{\cancel{3}}{\cancel{5}} \times \frac{\cancel{5}}{4} = -\frac{3}{4}$$ 6. **Interpret the result:** Since $\sin \theta$ is negative and $\cos \theta$ is positive, $\theta$ lies in the fourth quadrant where tangent is negative, confirming $\tan \theta = -\frac{3}{4}$. **Final answer:** $$\boxed{-\frac{3}{4}}$$