Find the value of `sin75^@`

Options

A. `(sqrt3+1)/(2sqrt2)`
B. `sqrt3/(2+sqrt2)`
C. `(sqrt3-1)/(2sqrt2)`
D. `(2+sqrt2)/sqrt3`

Answer

To find the value of \( \sin 75^\circ \), we can use the sine addition formula. The sine addition formula states that:

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

### Step-by-Step Solution:

1. **Express \( 75^\circ \) as a sum of two angles**:
\[ 75^\circ = 30^\circ + 45^\circ \]

2. **Apply the sine addition formula**:
\[ \sin 75^\circ = \sin(30^\circ + 45^\circ) = \sin 30^\circ \cos 45^\circ + \cos 30^\circ \sin 45^\circ \]

3. **Substitute the known values**:
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)

So we substitute these values into the equation:
\[ \sin 75^\circ = \left(\frac{1}{2} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}}\right) \]

4. **Simplify the expression**:
\[ \sin 75^\circ = \frac{1}{2\sqrt{2}} + \frac{\sqrt{3}}{2\sqrt{2}} \]

5. **Combine the fractions**:
\[ \sin 75^\circ = \frac{1 + \sqrt{3}}{2\sqrt{2}} \]

6. **Rationalize the denominator (optional)**:
To rationalize the denominator, multiply the numerator and the denominator by \( \sqrt{2} \):
\[ \sin 75^\circ = \frac{(1 + \sqrt{3})\sqrt{2}}{2 \cdot 2} = \frac{(1 + \sqrt{3})\sqrt{2}}{4} \]

### Final Answer:
\[ \sin 75^\circ = \frac{(1 + \sqrt{3})\sqrt{2}}{4} \]