Find the value of x, if
`sin 2 x = sin 60^(@) cos 30^(@) - cos 60^(@) sin 30^(@)`

Options

A. `20^(@)`
B. `15^(@)`
C. `30^(@)`
D. `45^(@)`

Answer

To solve the equation \( \sin 2x = \sin 60^\circ \cos 30^\circ - \cos 60^\circ \sin 30^\circ \), we will follow these steps:

### Step 1: Substitute the values of the trigonometric functions
We know the following values:
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \cos 60^\circ = \frac{1}{2} \)
- \( \sin 30^\circ = \frac{1}{2} \)

Substituting these values into the equation gives us:

\[ \sin 2x = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{1}{2}\right) \left(\frac{1}{2}\right) \]

### Step 2: Simplify the right-hand side
Calculating the right-hand side:

\[ \sin 2x = \frac{3}{4} - \frac{1}{4} \]

\[ \sin 2x = \frac{2}{4} = \frac{1}{2} \]

### Step 3: Set the equation for \( \sin 2x \)
Now we have:

\[ \sin 2x = \frac{1}{2} \]

### Step 4: Find the general solutions for \( 2x \)
The sine function equals \( \frac{1}{2} \) at the angles:

\[ 2x = 30^\circ + n \cdot 360^\circ \quad \text{or} \quad 2x = 150^\circ + n \cdot 360^\circ \]

where \( n \) is any integer.

### Step 5: Solve for \( x \)
Dividing both sides by 2 gives us:

1. \( x = 15^\circ + n \cdot 180^\circ \)
2. \( x = 75^\circ + n \cdot 180^\circ \)

### Step 6: Find the specific solution
For \( n = 0 \):

1. \( x = 15^\circ \)
2. \( x = 75^\circ \)

### Conclusion
Thus, the values of \( x \) that satisfy the equation are \( x = 15^\circ \) and \( x = 75^\circ \).