Question

Calculate the angle of 1’’ (second of arc or arc second) in radians.
(Use 360°=2π rad, 1°=60’ and 1’=60’’)

Answer 1

Hint: The given problem is from basic introduction to angles. The degree measurement has a relation with radian measurement of angle. So here first we convert 1’’ in degree form and then calculate its radian form.

Complete step by step solution:
As we know that the angle can be measured in degree as well as radian. There will be a relation for conversion of degree form into radian form that is given by
π radian = 180°
or
1° = π/180 radian
We can divide 1° angle into 60’ (minutes) then the 1’ (minute) can also be divided into 60’’ (seconds).
1° = 60’
1’ = 60’’
So according to degree and radian relation we can write 1° as
${60}^{\prime }={\displaystyle \frac{\pi }{180}}rad$
So, the value of 1’ in radian would be
$1^{\prime }={\displaystyle \frac{\pi }{180\times 60}}rad$
1’ (minute) is equal to 60’’ (seconds).
So the relation can be written as
${60}^{″}={\displaystyle \frac{\pi }{180\times 60}}rad$
The value of 1’’ will be
$$\begin{gathered} 1^{″}={\displaystyle \frac{\pi }{180\times 60\times 60}}rad \\ {1}^{″}={\displaystyle \frac{\pi }{648000}}radian \end{gathered}$$
This is the required solution for a given problem.
In geometry, an angle can be defined as the figure formed by two rays meeting at a common endpoint. An angle can be measured in degree and radian. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.

Note: Degree is a measure of an angle. One degree is one 360th part of a full circle. Each degree is divided into minutes and seconds. The degree is divided into 60 minutes. The minute is divided again into 60 seconds for fine measurements.