Value of `sin 15^@` is :

Options

A. `1`
B. `1//2`
C. `1//4`
D. `(sqrt(3))/(2)`

Explanation

To find the value of \( \sin 15^\circ \), we can use the sine subtraction formula. Here’s a step-by-step solution:

### Step 1: Express \( \sin 15^\circ \) using known angles
We can express \( 15^\circ \) as \( 45^\circ - 30^\circ \):
\[ \sin 15^\circ = \sin(45^\circ - 30^\circ) \]

### Step 2: Apply the sine subtraction formula
The sine subtraction formula states that:
\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]
Using this formula, we have:
\[ \sin 15^\circ = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ \]

### Step 3: Substitute the known values
We know the following trigonometric values:
- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sin 30^\circ = \frac{1}{2} \)

Now substituting these values into the equation:
\[ \sin 15^\circ = \left(\frac{1}{\sqrt{2}}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{1}{\sqrt{2}}\right) \left(\frac{1}{2}\right) \]

### Step 4: Simplify the expression
Calculating each term:
\[ \sin 15^\circ = \frac{\sqrt{3}}{2\sqrt{2}} - \frac{1}{2\sqrt{2}} \]
Now, combine the two fractions:
\[ \sin 15^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]

### Step 5: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{2} \):
\[ \sin 15^\circ = \frac{(\sqrt{3} - 1)\sqrt{2}}{2 \cdot 2} = \frac{\sqrt{6} - \sqrt{2}}{4} \]

### Conclusion
Thus, the value of \( \sin 15^\circ \) is:
\[ \sin 15^\circ = \frac{\sqrt{6} - \sqrt{2}}{4} \]