If each of `a, b and c` is a positive acute angle such that `sin (a + b - c) = 1//2, cos (b + c - a) = 1//2 and tan (c + a - b) = 1`, then find the values of `a, b and c` respectively.

Options

A. `30^(@), 60^(@), 90^(@)`
B. `45^(@) , 45^(@) , 90^(@)`
C. `30^(@), 45^(@), 60^(@)`
D. `(37 (1)/(2) )^(@) , 45^(@) , ( 52(1)/(2) )^(@)`

Answer

To solve the problem, we need to analyze the given equations step by step.

### Step 1: Set up the equations based on the trigonometric identities.
We have three equations based on the given conditions:
1. \( \sin(a + b - c) = \frac{1}{2} \)
2. \( \cos(b + c - a) = \frac{1}{2} \)
3. \( \tan(c + a - b) = 1 \)

### Step 2: Solve the first equation.
From \( \sin(a + b - c) = \frac{1}{2} \), we know that:
\[ a + b - c = 30^\circ \quad \text{(since } \sin 30^\circ = \frac{1}{2}\text{)} \]

### Step 3: Solve the second equation.
From \( \cos(b + c - a) = \frac{1}{2} \), we know that:
\[ b + c - a = 60^\circ \quad \text{(since } \cos 60^\circ = \frac{1}{2}\text{)} \]

### Step 4: Solve the third equation.
From \( \tan(c + a - b) = 1 \), we know that:
\[ c + a - b = 45^\circ \quad \text{(since } \tan 45^\circ = 1\text{)} \]

### Step 5: Write the system of equations.
Now we have the following system of equations:
1. \( a + b - c = 30^\circ \) (Equation 1)
2. \( b + c - a = 60^\circ \) (Equation 2)
3. \( c + a - b = 45^\circ \) (Equation 3)

### Step 6: Solve the equations.
Let's add Equation 1 and Equation 2:
\[ (a + b - c) + (b + c - a) = 30^\circ + 60^\circ \]
This simplifies to:
\[ 2b = 90^\circ \implies b = 45^\circ \]

### Step 7: Substitute \( b \) back into the equations.
Now substitute \( b = 45^\circ \) into Equation 1:
\[ a + 45^\circ - c = 30^\circ \implies a - c = 30^\circ - 45^\circ \implies a - c = -15^\circ \implies a = c - 15^\circ \quad \text{(Equation 4)} \]

Now substitute \( b = 45^\circ \) into Equation 2:
\[ 45^\circ + c - a = 60^\circ \implies c - a = 60^\circ - 45^\circ \implies c - a = 15^\circ \quad \text{(Equation 5)} \]

### Step 8: Solve Equations 4 and 5.
From Equation 4, we have \( a = c - 15^\circ \).
From Equation 5, we have \( c - a = 15^\circ \).

Substituting \( a \) from Equation 4 into Equation 5:
\[ c - (c - 15^\circ) = 15^\circ \implies 15^\circ = 15^\circ \]
This confirms our equations are consistent.

### Step 9: Find the values of \( a \) and \( c \).
Using \( a = c - 15^\circ \) and substituting \( c = a + 15^\circ \) into the third equation:
\[ c + a - 45^\circ = 45^\circ \implies (a + 15^\circ) + a - 45^\circ = 45^\circ \]
This simplifies to:
\[ 2a - 30^\circ = 45^\circ \implies 2a = 75^\circ \implies a = 37.5^\circ \]
Now substituting \( a \) back to find \( c \):
\[ c = a + 15^\circ = 37.5^\circ + 15^\circ = 52.5^\circ \]

### Final Values:
Thus, the values are:
\[ a = 37.5^\circ, \quad b = 45^\circ, \quad c = 52.5^\circ \]