If A,B,C the angles of a triangle be in A.P. and satisfy the relations
`sin(2A + B) = sin (C -A) = -sin(B + 2C) = 1/2`,
then the values of A,B,C are respectively

Options

A. `45^@, 75^@, 60^@`
B. `75^@, 60^@,45^@`
C. `60^@, 60^@, 60^@`
D. `45^@, 60^@, 75^@`

Answer

To solve the problem, we need to find the angles \( A, B, C \) of a triangle that are in Arithmetic Progression (A.P.) and satisfy the conditions given by the equations:

1. \( \sin(2A + B) = \frac{1}{2} \)
2. \( \sin(C - A) = \frac{1}{2} \)
3. \( -\sin(B + 2C) = \frac{1}{2} \)

Since \( A, B, C \) are angles of a triangle, we know that \( A + B + C = 180^\circ \).

### Step 1: Express B and C in terms of A
Since \( A, B, C \) are in A.P., we can express \( B \) and \( C \) as:
- \( B = A + d \)
- \( C = A + 2d \)

Where \( d \) is the common difference.

From the triangle angle sum property:
\[ A + (A + d) + (A + 2d) = 180^\circ \]
This simplifies to:
\[ 3A + 3d = 180^\circ \implies A + d = 60^\circ \]
Thus, we have:
\[ B = 60^\circ \quad \text{and} \quad C = 120^\circ - A \]

### Step 2: Solve the first equation
Using the first equation \( \sin(2A + B) = \frac{1}{2} \):
\[ \sin(2A + 60^\circ) = \frac{1}{2} \]
The angles for which \( \sin x = \frac{1}{2} \) are \( x = 30^\circ + n \cdot 360^\circ \) and \( x = 150^\circ + n \cdot 360^\circ \).

Setting \( 2A + 60^\circ = 30^\circ \):
\[ 2A = 30^\circ - 60^\circ \implies 2A = -30^\circ \quad \text{(not possible)} \]

Setting \( 2A + 60^\circ = 150^\circ \):
\[ 2A = 150^\circ - 60^\circ \implies 2A = 90^\circ \implies A = 45^\circ \]

### Step 3: Find B and C
Now substituting \( A = 45^\circ \) back into the expressions for \( B \) and \( C \):
\[ B = 60^\circ \quad \text{and} \quad C = 120^\circ - 45^\circ = 75^\circ \]

### Step 4: Verify with the second equation
Now we check the second equation \( \sin(C - A) = \frac{1}{2} \):
\[ C - A = 75^\circ - 45^\circ = 30^\circ \]
\[ \sin(30^\circ) = \frac{1}{2} \quad \text{(True)} \]

### Step 5: Verify with the third equation
Now check the third equation \( -\sin(B + 2C) = \frac{1}{2} \):
\[ B + 2C = 60^\circ + 2 \times 75^\circ = 60^\circ + 150^\circ = 210^\circ \]
\[ -\sin(210^\circ) = -(-\frac{1}{2}) = \frac{1}{2} \quad \text{(True)} \]

### Conclusion
Thus, the values of \( A, B, C \) are:
\[ A = 45^\circ, \quad B = 60^\circ, \quad C = 75^\circ \]