1. **State the problem:** Find the exact value of $\sin(105^\circ)$. 2. **Use the sum formula for sine:** The angle $105^\circ$ can be expressed as $60^\circ + 45^\circ$. The sine sum identity is $$\sin(a+b) = \sin a \cos b + \cos a \sin b$$ 3. **Apply the formula:** $$\sin(105^\circ) = \sin(60^\circ + 45^\circ) = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ$$ 4. **Recall exact values:** $$\sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \cos 60^\circ = \frac{1}{2}, \quad \sin 45^\circ = \frac{\sqrt{2}}{2}$$ 5. **Substitute values:** $$\sin(105^\circ) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$$ 6. **Combine terms:** $$\sin(105^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}$$ **Final answer:** $$\sin(105^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}$$