Question

Express in radian the following angle:
(a)$175{}^\circ 45'$
(b)$47{}^\circ 25'36''$

Answer 1

Hint: First we will use the minute of arc method to convert minute into degree in decimals and second of arc method to convert second to 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 will use the second of the arc methods to form the relation between degree and second.
A second of arc is $\dfrac{1}{60}$ of arcminute which is $\dfrac{1}{3600}$ of $1{}^\circ$.
Therefore, from this we get $3600\sec =1{}^\circ .............(2)$
Hence the degree has been converted to minutes and second, and we have the result that we need to solve this question.
For (a):
Now we have$175{}^\circ 45'$, so we have to convert 45 minute to degrees.
By using the relation (1) we get,
$\dfrac{45}{60}=\dfrac{45}{60}$
45 minute = 0.75 degree
Now we use this to convert our whole question into degree,
Hence, $175{}^\circ 45'=175{}^\circ +0.75{}^\circ =175.75{}^\circ$
Now we will use the formula that helps us to convert degrees 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 175.75 in the above formula we get,
$\begin{aligned} & \text{Angle in radian}=\dfrac{\pi }{180}\times 175.75 \\\ & \text{Angle in radian}=\dfrac{703\pi }{720} \\\ \end{aligned}$
Hence, the answer for (a) is $\dfrac{703\pi }{720}$ .
For (b):
Now we have $47{}^\circ 25'36''$, so we have to convert 25 minute to degrees.
By using the relation (1) we get,
$\dfrac{25}{60}=\dfrac{25}{60}$
25 minute = 0.417 degree
Now we will convert 36 second to degrees.
By using relation (2) we get,
36 second = 0.01 degree
Now we use this to convert our whole question into degree,
Hence, $47{}^\circ 25'36''=47{}^\circ +0.417{}^\circ +0.01{}^\circ =47.427{}^\circ$
Now we will use the formula that helps us to convert degrees 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 47.427 in the above formula we get,
$\begin{aligned} & \text{Angle in radian}=\dfrac{\pi }{180}\times 47.427 \\\ & \text{Angle in radian}=\dfrac{15809\pi }{60000} \\\ \end{aligned}$
Hence, the answer for (b) is $\dfrac{15809\pi }{60000}$ .

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.