Find the value of: `sin75^(0)`

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 that:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
Using \( a = 45^\circ \) and \( b = 30^\circ \), we have:
\[ \sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \]

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

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

### Step 4: Simplify the expression
Now we simplify the expression:
\[ \sin 75^\circ = \frac{1 \cdot \sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} \]
Combining the terms:
\[ \sin 75^\circ = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

### Step 5: Final answer
Thus, the value of \( \sin 75^\circ \) is:
\[ \sin 75^\circ = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]