If the sum and difference of two angles are ` (22)/(9)` radian and `36^(@)` respectively, then the value of smaller angle in degree taking the value of` pi` as as `(22)/(7)` is:

Options

A. (a)`52 ^(@)`
B. (b)`48^(@) `
C. (c)`56 ^(@)`
D. (d)`60^(@)`

Explanation

To solve the problem step by step, we need to find the smaller angle given the sum and difference of two angles. Here's how we can do it:

### Step 1: Convert the sum from radians to degrees
The sum of the two angles is given as \( \frac{22}{9} \) radians. We know that \( \pi \) radians is equal to \( 180^\circ \).

To convert radians to degrees, we use the formula:
\[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]

Substituting \( \pi \) with \( \frac{22}{7} \):
\[ \text{Degrees} = \frac{22}{9} \times \frac{180}{\frac{22}{7}} = \frac{22}{9} \times \frac{180 \times 7}{22} \]

The \( 22 \) cancels out:
\[ = \frac{180 \times 7}{9} = \frac{1260}{9} = 140^\circ \]

### Step 2: Set up the equations for the angles
Let the two angles be \( A \) and \( B \). We know:
1. \( A + B = 140^\circ \) (Equation 1)
2. \( A - B = 36^\circ \) (Equation 2)

### Step 3: Solve the equations
To find \( A \) and \( B \), we can add Equation 1 and Equation 2:
\[ (A + B) + (A - B) = 140^\circ + 36^\circ \]
This simplifies to:
\[ 2A = 176^\circ \]
Dividing both sides by 2:
\[ A = 88^\circ \]

### Step 4: Find the value of \( B \)
Now, substitute \( A \) back into Equation 1:
\[ 88^\circ + B = 140^\circ \]
Solving for \( B \):
\[ B = 140^\circ - 88^\circ = 52^\circ \]

### Step 5: Identify the smaller angle
Now we have both angles:
- \( A = 88^\circ \)
- \( B = 52^\circ \)

The smaller angle is:
\[ \text{Smaller angle} = 52^\circ \]

### Final Answer
The value of the smaller angle is \( 52^\circ \).

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