Question

Express in radian the given angle: $110{}^\circ 30'$

Answer 1

Hint: First we will use the minute of arc method to convert the 30 minute into degree in decimals and then we will use the formula $\text{Angle in radian}=\dfrac{\pi }{180}\times \text{Angle in degree}$ , to convert degree into radian and then we will substitute the value of degree in the formula that is given in the question to find the value in radian and that will be the final answer.

Complete step-by-step answer:
Let’s start our solution by first writing the relation between degree to minutes and seconds.
First we will use the minute of arc method for the relation between degree and minute.
A minute of arc or arcminute is a unit of angular measurement equal to $\dfrac{1}{60}$ of $1{}^\circ$ .
Therefore, from this we get $60\min =1{}^\circ .............(1)$
Now we have $110{}^\circ 30'$, so we have to convert 30 minute to degree.
By using the relation (1) we get,
30 minute = 0.5 degree
Now we use this to convert our whole question into degree,
Hence, $110{}^\circ 30'=110{}^\circ +0.5{}^\circ =110.5{}^\circ$
Now we will use the formula that helps us to convert degree into radian.
The formula that converts degree into radian is:
$\text{Angle in radian}=\dfrac{\pi }{180}\times \text{Angle in degree}$
Now substituting the value of degree as 110.5 in the above formula we get,
$\begin{aligned} & \text{Angle in radian}=\dfrac{\pi }{180}\times 110.5 \\\ & \text{Angle in radian}=\dfrac{221\pi }{360} \\\ \end{aligned}$
Hence, the answer to this question is $\dfrac{221\pi }{360}$ .

Note: The formula that we have used for conversion is$\text{Angle in radian}=\dfrac{\pi }{180}\times \text{Angle in degree}$ and the minute of arc method must be kept in mind. And this formula must be kept in mind. From this formula we can also find the value of angle in degree if the value of angle in radian is given. So, in some questions the value of angle in radian might be given and we need to find the value of angle in degree, then also we will use the same formula for the purpose. We can also use the fact that 1 degree = 0.0174 radian, and then we can multiply it by 60 to get the value of 60 degree in radian.