If ` sqrt2 sin (60^(@) - theta) = 1,` then the value of ` theta ` is

Options

A. `20^(@)`
B. `30^(@)`
C. `15 ^(@)`
D. None of these

Answer

To solve the equation \( \sqrt{2} \sin(60^\circ - \theta) = 1 \), we will follow these steps:

### Step 1: Isolate the sine function
We start with the given equation:
\[ \sqrt{2} \sin(60^\circ - \theta) = 1 \]
To isolate the sine function, we divide both sides by \( \sqrt{2} \):
\[ \sin(60^\circ - \theta) = \frac{1}{\sqrt{2}} \]

### Step 2: Recognize the sine value
We know that \( \frac{1}{\sqrt{2}} \) is equal to \( \sin(45^\circ) \). Therefore, we can rewrite the equation as:
\[ \sin(60^\circ - \theta) = \sin(45^\circ) \]

### Step 3: Set up the angle equations
Since the sine function is equal for angles that differ by \( n \cdot 180^\circ \) or are supplementary, we can set up the following equations:
1. \( 60^\circ - \theta = 45^\circ + n \cdot 180^\circ \)
2. \( 60^\circ - \theta = 180^\circ - 45^\circ + n \cdot 180^\circ \)

For simplicity, we will consider \( n = 0 \) first.

### Step 4: Solve the first equation
From the first equation:
\[ 60^\circ - \theta = 45^\circ \]
Rearranging gives:
\[ \theta = 60^\circ - 45^\circ = 15^\circ \]

### Step 5: Solve the second equation
Now, let's solve the second equation:
\[ 60^\circ - \theta = 135^\circ \]
Rearranging gives:
\[ \theta = 60^\circ - 135^\circ = -75^\circ \]
Since \( \theta \) must be a positive angle in this context, we discard this solution.

### Conclusion
Thus, the only valid solution is:
\[ \theta = 15^\circ \]