Find the value of: `sin75^(@)`

Options

A. 0
B. `(sqrt1)/(2sqrt(2))`
C. `sqrt2`
D. `(sqrt(3)+1)/(2sqrt(2))`

Explanation

To find the value of \( \sin 75^\circ \), we can use the sine addition formula. Here’s a step-by-step solution:

### Step 1: Express \( 75^\circ \) as a sum of two angles
We can express \( 75^\circ \) as:
\[ 75^\circ = 45^\circ + 30^\circ \]

### Step 2: Apply the sine addition formula
The sine addition formula states:
\[ \sin(x + y) = \sin x \cos y + \cos x \sin y \]
Applying this to our angles:
\[ \sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \]

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

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

### Step 4: Simplify the expression
Calculating each term:
1. \( \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{4} \)
2. \( \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{4} \)

Now combine the two results:
\[ \sin 75^\circ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \]

### Final Answer
Thus, the value of \( \sin 75^\circ \) is:
\[ \sin 75^\circ = \frac{\sqrt{6} + \sqrt{2}}{4} \]