Radian measure of `40^(@)20'` is equal to

Options

A. `(120pi)/(504) ` radian
B. `(121pi)/(540) ` radian
C. `(121pi)/(3)` radian
D. None of these

Answer

To convert the angle \(40^\circ 20'\) into radians, we will follow these steps:

### Step 1: Convert minutes to degrees
We know that:
- \(1 \text{ degree} = 60 \text{ minutes}\)

Thus, to convert \(20\) minutes into degrees:
\[ 20' = 20 \times \frac{1}{60} \text{ degrees} = \frac{20}{60} \text{ degrees} = \frac{1}{3} \text{ degrees} \]

### Step 2: Add the degrees and converted minutes
Now, we can add the degrees and the converted minutes:
\[ 40^\circ 20' = 40^\circ + \frac{1}{3}^\circ = 40 + \frac{1}{3} = \frac{120}{3} + \frac{1}{3} = \frac{121}{3} \text{ degrees} \]

### Step 3: Convert degrees to radians
We know the conversion factor between degrees and radians:
\[ 180^\circ = \pi \text{ radians} \]
Thus, \(1 \text{ degree} = \frac{\pi}{180} \text{ radians}\).

Now we convert \(\frac{121}{3} \text{ degrees}\) to radians:
\[ \frac{121}{3} \text{ degrees} = \frac{121}{3} \times \frac{\pi}{180} \text{ radians} \]

### Step 4: Simplify the expression
Now we can simplify the expression:
\[ \frac{121 \pi}{3 \times 180} = \frac{121 \pi}{540} \text{ radians} \]

### Final Answer
Thus, the radian measure of \(40^\circ 20'\) is:
\[ \frac{121 \pi}{540} \text{ radians} \]

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