Question

Calculate the angle of 1’ (minute of arc or arc min).

Answer 1

Hint:
We will convert $$\pi$$ into degrees for getting angle for 1’ and we will see what the arc minute is. We will convert the radian into arc minute. Then we will find the radian of the arc min, then we will convert it into the angle into degree. We will form an equation and solve further to find the solution; this will provide us the answer.

Complete step by step solution:
As we know, $$\pi$$ radian $$={180}^{\circ }$$
By the above we can say that
  $$1^{\circ }={\displaystyle \frac{\pi }{{180}^{\circ }}}$$ radians …(1)
As we will convert degree into the minutes
 $$\Rightarrow 1^{\circ }={60}^{\prime }$$ (‘ indicates minutes)
So, by equation 1 we can say that
 $$\Rightarrow 1^{\circ }={\displaystyle \frac{\pi }{{180}^{\circ }}}$$
We will replace $$1^{\circ }$$ to 60’ in the equation 1
 $$\Rightarrow {60}^{\prime }={\displaystyle \frac{\pi }{{180}^{\circ }}}$$
As we want to find the 1’ so we will the equation by 60,
  $$\Rightarrow 1^{\prime }={\displaystyle \frac{\pi }{(180\times 60)}}$$ radian
After calculating we will get
 $$\Rightarrow 1^{\prime }={\displaystyle \frac{\pi }{10800}}$$ radian
Therefore the 1’ is
 $$\Rightarrow 1^{\prime }={\displaystyle \frac{\pi }{10800}}$$ radian
As we know the radian to degree conversion formula is
 $$\Rightarrow min=\left(x\times {\displaystyle \frac{180}{\pi }}\right)\mathrm{deg}$$
Now we have the value of x, we will put in the above conversion equation
 $$\Rightarrow 1^{\prime }={\displaystyle \frac{\pi }{10800}}\times {\displaystyle \frac{180}{\pi }}$$

Hence on simplification we get,
 $$\Rightarrow 1^{\prime }={0.0167}^{\circ }$$

Note:
A minute of arc, arc minute, or minute arc is a unit of angular measurement equal to one degree. Since one degree is of a turn, one minute of arc is of a turn. A minute of arc is of a radian. It is used to find the degree into time form, which is needed in terms of finding time and also this conversion helps in finding the planet's latitude and longitude to find the exact location.