1. **Problem:** Find the exact value of $\sin(225^\circ)$. 2. **Formula and rules:** The sine of an angle in the unit circle can be found using reference angles and the signs in each quadrant. Since $225^\circ$ is in the third quadrant, where sine is negative, and its reference angle is $225^\circ - 180^\circ = 45^\circ$, we use: $$\sin(225^\circ) = -\sin(45^\circ)$$ Recall that $\sin(45^\circ) = \frac{\sqrt{2}}{2}$. 3. **Calculation:** $$\sin(225^\circ) = -\frac{\sqrt{2}}{2}$$ 4. **Explanation:** The angle $225^\circ$ lies in the third quadrant where sine values are negative. The reference angle is $45^\circ$, so the sine value is the negative of $\sin(45^\circ)$. **Final answer:** $$\boxed{-\frac{\sqrt{2}}{2}}$$