Questions: Use the half-angle formulas to find the exact value of the trigonometric function sin 22.5°. sin 22.5° = (Type an exact answer, using radicals as needed.)

Solution Steps

To find \( \sin 22.5^{\circ} \), we recognize that \( 22.5^{\circ} \) is half of \( 45^{\circ} \): \[ 22.5^{\circ} = \frac{45^{\circ}}{2} \]

We calculate the cosine of \( 45^{\circ} \): \[ \cos 45^{\circ} = \frac{\sqrt{2}}{2} \approx 0.7071 \]

Using the half-angle formula for sine: \[ \sin \left( \frac{\theta}{2} \right) = \sqrt{\frac{1 - \cos \theta}{2}} \] we substitute \( \theta = 45^{\circ} \): \[ \sin 22.5^{\circ} = \sqrt{\frac{1 - \cos 45^{\circ}}{2}} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \]

Now we simplify the expression: \[ \sin 22.5^{\circ} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2} \]

Final Answer