What is the value of sin `15^(@)` ?

Options

A. `(sqrt(3)-1)/(2sqrt(2))`
B. `(sqrt(3)+1)/(2sqrt(2))`
C. `(sqrt(3)-1)/(sqrt(3)+1)`
D. `(sqrt(3)+1)/(sqrt(3)-1)`

Explanation

To find the value of \( \sin 15^\circ \), we can use the sine subtraction formula. The angle \( 15^\circ \) can be expressed as \( 45^\circ - 30^\circ \).

### Step-by-step Solution:

1. **Identify the Angles**:
\[ \sin 15^\circ = \sin(45^\circ - 30^\circ) \]

2. **Apply the Sine Subtraction Formula**:
The sine subtraction formula states that:
\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]
Here, \( a = 45^\circ \) and \( b = 30^\circ \).

3. **Substitute the Values**:
\[ \sin 15^\circ = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ \]

4. **Use Known Values**:
We know:
\[ \sin 45^\circ = \frac{1}{\sqrt{2}}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \cos 45^\circ = \frac{1}{\sqrt{2}}, \quad \sin 30^\circ = \frac{1}{2} \]

5. **Substitute the Known Values**:
\[ \sin 15^\circ = \left(\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}\right) - \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{2}\right) \]

6. **Simplify the Expression**:
\[ \sin 15^\circ = \frac{\sqrt{3}}{2\sqrt{2}} - \frac{1}{2\sqrt{2}} \]
Combine the fractions:
\[ \sin 15^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]

### Final Answer:
\[ \sin 15^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]