Using the half angle formulas, find the exact value of `(i) sin 15 ^(@) (ii) sin 22 (1)/(2) ""^(@).`

Answer

To find the exact values of \( \sin 15^\circ \) and \( \sin 22.5^\circ \) using half angle formulas, we can follow these steps:

### Step-by-Step Solution

#### Part (i): Finding \( \sin 15^\circ \)

1. **Identify the angle**: We can express \( 15^\circ \) as \( \frac{30^\circ}{2} \).

2. **Use the half angle formula**: The half angle formula for sine is given by:
\[ \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \]
Here, \( \theta = 30^\circ \).

3. **Calculate \( \cos 30^\circ \)**: We know that:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]

4. **Substitute into the half angle formula**:
\[ \sin 15^\circ = \sin \frac{30^\circ}{2} = \sqrt{\frac{1 - \cos 30^\circ}{2}} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \]

5. **Simplify the expression**:
\[ \sin 15^\circ = \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{3}}{4}} = \frac{\sqrt{2 - \sqrt{3}}}{2} \]

#### Part (ii): Finding \( \sin 22.5^\circ \)

1. **Identify the angle**: We can express \( 22.5^\circ \) as \( \frac{45^\circ}{2} \).

2. **Use the half angle formula**: Again using the half angle formula for sine:
\[ \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \]
Here, \( \theta = 45^\circ \).

3. **Calculate \( \cos 45^\circ \)**: We know that:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

4. **Substitute into the half angle formula**:
\[ \sin 22.5^\circ = \sin \frac{45^\circ}{2} = \sqrt{\frac{1 - \cos 45^\circ}{2}} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \]

5. **Simplify the expression**:
\[ \sin 22.5^\circ = \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2} \]

### Final Answers
- \( \sin 15^\circ = \frac{\sqrt{2 - \sqrt{3}}}{2} \)
- \( \sin 22.5^\circ = \frac{\sqrt{2 - \sqrt{2}}}{2} \)