Express in radian the fourth angle of a quadrilateral which has three angle `46^(@) 30' 10''`, `75^(@) 44' 45'' ` and `123^(@) 9' 35'' ` respectively, taking `pi=(355)/(113)`

Answer

To find the fourth angle of a quadrilateral given the other three angles, and then convert that angle into radians, we can follow these steps:

### Step 1: Identify the given angles
The three angles of the quadrilateral are:
1. \( 46^\circ 30' 10'' \)
2. \( 75^\circ 44' 45'' \)
3. \( 123^\circ 9' 35'' \)

### Step 2: Convert all angles to seconds
To simplify the addition, we convert each angle to seconds (since \(1^\circ = 3600''\)):
- For \( 46^\circ 30' 10'' \):
\[ 46^\circ = 46 \times 3600'' = 165600'' \]
\[ 30' = 30 \times 60'' = 1800'' \]
\[ 10'' = 10'' \]
Total:
\[ 165600'' + 1800'' + 10'' = 167410'' \]

- For \( 75^\circ 44' 45'' \):
\[ 75^\circ = 75 \times 3600'' = 270000'' \]
\[ 44' = 44 \times 60'' = 2640'' \]
\[ 45'' = 45'' \]
Total:
\[ 270000'' + 2640'' + 45'' = 272685'' \]

- For \( 123^\circ 9' 35'' \):
\[ 123^\circ = 123 \times 3600'' = 442800'' \]
\[ 9' = 9 \times 60'' = 540'' \]
\[ 35'' = 35'' \]
Total:
\[ 442800'' + 540'' + 35'' = 443375'' \]

### Step 3: Sum the three angles in seconds
Now, we add the three angles together:
\[ 167410'' + 272685'' + 443375'' = 883470'' \]

### Step 4: Calculate the fourth angle
The sum of angles in a quadrilateral is \(360^\circ\), which is:
\[ 360^\circ = 360 \times 3600'' = 1296000'' \]
Thus, the fourth angle in seconds is:
\[ 1296000'' - 883470'' = 412530'' \]

### Step 5: Convert the fourth angle back to degrees
To convert \(412530''\) back to degrees:
1. Find the degrees:
\[ \text{Degrees} = \frac{412530''}{3600''} = 114.025 \]
This means:
\[ 114^\circ + 0.025 \times 60' = 114^\circ + 1.5' = 114^\circ 1.5' \]
Further converting \(0.5'\) to seconds:
\[ 0.5' = 0.5 \times 60'' = 30'' \]
So, the fourth angle is:
\[ 114^\circ 1' 30'' \]

### Step 6: Convert the fourth angle to radians
Using the conversion factor \( \frac{\pi}{180} \):
\[ \text{Fourth angle in radians} = 114.025 \times \frac{\pi}{180} \]
Substituting \( \pi = \frac{355}{113} \):
\[ = 114.025 \times \frac{355}{113 \times 180} \]
Calculating:
\[ = \frac{114.025 \times 355}{20340} \]
This results in approximately \(2\) radians.

### Final Answer
The fourth angle of the quadrilateral in radians is approximately \(2\) radians.

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