Use the half-angle formulas to find the exact value of the trigonometric function \( \sin 22.5^{\circ} \). \( \sin 22.5^{\circ}=\square \) (Type an exact answer, using radicals as needed.)

To find the exact value of \( \sin 22.5^{\circ} \) using the half-angle formula, follow these steps: 1. **Recognize the Angle Relationship:** \( 22.5^{\circ} \) is half of \( 45^{\circ} \), so we can use the half-angle formula for sine: \[ \sin \left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{2}} \] Here, \( \theta = 45^{\circ} \). 2. **Apply the Half-Angle Formula:** \[ \sin 22.5^{\circ} = \sqrt{\frac{1 - \cos 45^{\circ}}{2}} \] 3. **Substitute the Value of \( \cos 45^{\circ} \):** \[ \cos 45^{\circ} = \frac{\sqrt{2}}{2} \] So, \[ \sin 22.5^{\circ} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \] 4. **Simplify the Expression:** \[ \sin 22.5^{\circ} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2} \] **Final Answer:** \[ \sin 22.5^{\circ} = \frac{\sqrt{2 - \sqrt{2}}}{2} \]