1. **State the problem:** Find the exact area swept by a windshield wiper of length $1.5$ m moving through an angle of $75^\circ$. 2. **Formula for area of a sector:** The area $A$ of a sector with radius $r$ and angle $\theta$ (in radians) is: $$A = \frac{1}{2} r^2 \theta$$ 3. **Convert angle to radians:** $$\theta = 75^\circ \times \frac{\pi}{180^\circ} = \frac{75\pi}{180} = \frac{5\pi}{12}$$ 4. **Substitute values:** $$A = \frac{1}{2} \times (1.5)^2 \times \frac{5\pi}{12}$$ 5. **Calculate intermediate steps:** $$A = \frac{1}{2} \times 2.25 \times \frac{5\pi}{12} = \frac{2.25}{2} \times \frac{5\pi}{12} = 1.125 \times \frac{5\pi}{12}$$ 6. **Simplify multiplication:** $$A = \frac{1.125 \times 5\pi}{12} = \frac{5.625\pi}{12}$$ 7. **Final exact answer:** $$\boxed{\frac{5.625\pi}{12} \text{ m}^2}$$ This is the exact area swept by the wiper.