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. The sine subtraction formula states that:

\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]

In this case, we can express \( 15^\circ \) as \( 45^\circ - 30^\circ \). Therefore, we have:

\[ \sin 15^\circ = \sin(45^\circ - 30^\circ) \]

Now, applying the sine subtraction formula:

\[ \sin 15^\circ = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ \]

Next, we need to substitute the known values of the sine and cosine for the angles involved:

- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sin 30^\circ = \frac{1}{2} \)

Now substituting these values into the equation:

\[ \sin 15^\circ = \left(\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}\right) - \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{2}\right) \]

This simplifies to:

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

Now, we can combine the two fractions:

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

To express this in a more simplified form, we can rationalize the denominator:

\[ \sin 15^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{(\sqrt{3} - 1)\sqrt{2}}{4} \]

Now, we can approximate the values:

- \( \sqrt{3} \approx 1.732 \)
- \( \sqrt{2} \approx 1.414 \)

Substituting these approximations:

\[ \sin 15^\circ \approx \frac{(1.732 - 1) \cdot 1.414}{4} = \frac{0.732 \cdot 1.414}{4} \]

Calculating this gives:

\[ \sin 15^\circ \approx \frac{1.033}{4} \approx 0.25825 \]

Thus, the value of \( \sin 15^\circ \) is approximately \( 0.258 \).