Value of $40^{\circ}20^{\prime}$ in Radian is

Question

Accepted Answer

To convert an angle from degrees to radians, we use the conversion factor $\frac{\pi}{180}$. The given angle is $40^{\circ}20^{\prime}$, which can be expressed in decimal degrees as $40 + \frac{20}{60} = 40.3333$ degrees.

Now, let's convert this angle to radians:

$$
40.3333 \times \frac{\pi}{180} = \frac{40.3333 \times \pi}{180}
$$

To simplify this, we multiply:

$$
40.3333 \times \pi \approx 126.9999\pi
$$

Now, we simplify the fraction:

$$
\frac{126.9999\pi}{180} \approx \frac{121\pi}{540}
$$

Thus, the value of $40^{\circ}20^{\prime}$ in radians is approximately $\frac{121\pi}{540}$, which corresponds to Option 1.

### Explanation for Other Options:

- **Option 2: $\frac{540\pi}{121}$**

This option suggests a conversion that is not consistent with the correct conversion factor from degrees to radians. The numerator and denominator are swapped compared to the correct answer, leading to an incorrect value.

- **Option 3: $\frac{121}{540\pi}$**

This option incorrectly places $\pi$ in the denominator, which is not how radians are expressed. Radians are typically expressed with $\pi$ in the numerator when converting from degrees.

- **Option 4: $\frac{540}{121\pi}$**

Similar to Option 3, this option incorrectly places $\pi$ in the denominator and reverses the correct ratio, leading to an incorrect expression for radians.

In conclusion, Option 1 is the correct answer because it accurately represents the conversion of $40^{\circ}20^{\prime}$ into radians using the standard conversion factor. The other options fail to correctly apply this conversion, either by misplacing $\pi$ or by incorrectly arranging the ratio.