If `sin (3x - 20^@) = cos(3y + 20^@)`, then the value of (x + y) is

Options

A. `20^(@)`
B. `30^(@)`
C. `40^(@)`
D. `45^(@)`

Answer

To solve the equation \( \sin(3x - 20^\circ) = \cos(3y + 20^\circ) \), we can use the property that states:

\[ \sin(A) = \cos(B) \implies A + B = 90^\circ \]

### Step-by-step Solution:

1. **Identify A and B**:
- Let \( A = 3x - 20^\circ \)
- Let \( B = 3y + 20^\circ \)

2. **Set up the equation**:
- According to the property, we have:
\[ (3x - 20^\circ) + (3y + 20^\circ) = 90^\circ \]

3. **Simplify the equation**:
- Combine like terms:
\[ 3x - 20^\circ + 3y + 20^\circ = 90^\circ \]
- The \( -20^\circ \) and \( +20^\circ \) cancel each other out:
\[ 3x + 3y = 90^\circ \]

4. **Factor out the common term**:
- Factor out 3 from the left side:
\[ 3(x + y) = 90^\circ \]

5. **Solve for \( x + y \)**:
- Divide both sides by 3:
\[ x + y = \frac{90^\circ}{3} = 30^\circ \]

### Final Answer:
The value of \( x + y \) is \( 30^\circ \).