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

Options

A. `52^(@)`
B. `60^(@)`
C. `56^(@)`
D. `48^(@)`

Answer

To find the smaller angle given the sum and difference of two angles, we can follow these steps:

### Step 1: Convert the sum of angles from radians to degrees
The sum of the two angles is given as \( \frac{22}{9} \) radians. To convert this to degrees, we use the formula:

\[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]

Given \( \pi = \frac{22}{7} \), we can substitute:

\[ \text{Degrees} = \frac{22}{9} \times \frac{180}{\frac{22}{7}} \]

### Step 2: Simplify the expression
We can simplify the expression:

\[ \text{Degrees} = \frac{22}{9} \times \frac{180 \times 7}{22} \]

The \( 22 \) cancels out:

\[ \text{Degrees} = \frac{180 \times 7}{9} \]

Now, simplify \( \frac{180}{9} = 20 \):

\[ \text{Degrees} = 20 \times 7 = 140 \text{ degrees} \]

### Step 3: Set up equations for the angles
Let the two angles be \( A \) and \( B \). We have:

1. \( A + B = 140 \) degrees (sum of angles)
2. \( A - B = 36 \) degrees (difference of angles)

### Step 4: Solve the equations
We can add the two equations:

\[ (A + B) + (A - B) = 140 + 36 \]

This simplifies to:

\[ 2A = 176 \]

Dividing both sides by 2 gives:

\[ A = 88 \text{ degrees} \]

### Step 5: Find the value of \( B \)
Now, substitute \( A \) back into one of the equations to find \( B \):

\[ A + B = 140 \]

Substituting \( A = 88 \):

\[ 88 + B = 140 \]

Subtracting 88 from both sides gives:

\[ B = 52 \text{ degrees} \]

### Step 6: Identify the smaller angle
The smaller angle between \( A \) and \( B \) is:

\[ \text{Smaller angle} = 52 \text{ degrees} \]

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

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