The value of `sin22(1)/(2)` will be

Options

A. `sqrt(2)-1`
B. `(sqrt(2)+1)/(2sqrt(2))`
C. `(sqrt(2)-1)/(sqrt(2))`
D. `sqrt((sqrt(2)-1)/(2sqrt(2)))`

Answer

To find the value of \( \sin(22.5^\circ) \), we can use the half-angle identity for sine. Here’s a step-by-step solution:

### Step 1: Recognize the angle
We know that \( 22.5^\circ \) is half of \( 45^\circ \). Therefore, we can express \( \sin(22.5^\circ) \) as:
\[ \sin(22.5^\circ) = \sin\left(\frac{45^\circ}{2}\right) \]

### Step 2: Use the half-angle formula
The half-angle formula for sine is given by:
\[ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}} \]
In our case, \( \theta = 45^\circ \). Thus, we can write:
\[ \sin(22.5^\circ) = \sqrt{\frac{1 - \cos(45^\circ)}{2}} \]

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

### Step 4: Substitute \( \cos(45^\circ) \) into the formula
Now substituting \( \cos(45^\circ) \) into our half-angle formula:
\[ \sin(22.5^\circ) = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \]

### Step 5: Simplify the expression
First, simplify the numerator:
\[ 1 - \frac{\sqrt{2}}{2} = \frac{2}{2} - \frac{\sqrt{2}}{2} = \frac{2 - \sqrt{2}}{2} \]
Now substitute this back into the sine expression:
\[ \sin(22.5^\circ) = \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2} \]

### Step 6: Final expression
Thus, we have:
\[ \sin(22.5^\circ) = \frac{\sqrt{2 - \sqrt{2}}}{2} \]

### Conclusion
The value of \( \sin(22.5^\circ) \) can also be expressed as:
\[ \sin(22.5^\circ) = \frac{\sqrt{2 - \sqrt{2}}}{2} \]