Convert it `63^(@) 14'51"` into radian .

Answer

To convert the angle \( 63^\circ 14' 51'' \) into radians, we will follow these steps:

### Step 1: Convert minutes and seconds to degrees
1. **Convert minutes to degrees**:
\[ 14' = \frac{14}{60} \text{ degrees} \]
2. **Convert seconds to degrees**:
\[ 51'' = \frac{51}{3600} \text{ degrees} \]

### Step 2: Combine all parts into degrees
Now, we can express the entire angle in degrees:
\[ \text{Total degrees} = 63 + \frac{14}{60} + \frac{51}{3600} \]

### Step 3: Find a common denominator
The common denominator for \( 60 \) and \( 3600 \) is \( 3600 \). We will convert each term:
1. **Convert \( 63 \) degrees**:
\[ 63 = \frac{63 \times 3600}{3600} = \frac{226800}{3600} \]
2. **Convert \( \frac{14}{60} \) degrees**:
\[ \frac{14}{60} = \frac{14 \times 60}{3600} = \frac{840}{3600} \]
3. **Convert \( \frac{51}{3600} \) degrees**:
\[ \frac{51}{3600} = \frac{51}{3600} \]

### Step 4: Add all parts together
Now, we can add them:
\[ \text{Total degrees} = \frac{226800 + 840 + 51}{3600} = \frac{227691}{3600} \]

### Step 5: Convert degrees to radians
To convert degrees to radians, we use the formula:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
Substituting our total degrees:
\[ \text{radians} = \frac{227691}{3600} \times \frac{\pi}{180} \]

### Step 6: Simplify the expression
1. **Multiply the fractions**:
\[ \text{radians} = \frac{227691 \pi}{3600 \times 180} \]
2. **Calculate \( 3600 \times 180 \)**:
\[ 3600 \times 180 = 648000 \]
3. **Final expression**:
\[ \text{radians} = \frac{227691 \pi}{648000} \]

### Step 7: Simplify further by finding common factors
To simplify \( \frac{227691}{648000} \):
1. **Find GCD**: The GCD of \( 227691 \) and \( 648000 \) is \( 9 \).
2. **Divide both by \( 9 \)**:
\[ \frac{227691 \div 9}{648000 \div 9} = \frac{25299 \pi}{72000} \]

Thus, the final answer is:
\[ \text{radians} = \frac{25299 \pi}{72000} \]