Question

Express in degrees, minutes and seconds the given angle: $10\pi {}^\circ$

Answer 1

Hint: First we will convert $10\pi {}^\circ$ into decimal and then we will write how many minutes and second are there in one degree, and then we will look at the given expression, and if there is no decimal part then it has only degrees, no minutes and second. And if it has a decimal part then the number before decimal is the degree and for the number after decimal we will use the relation between degree minutes and second to convert the decimal into minutes and second.

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.
Now we have given $10\pi {}^\circ$ and we have to convert it into decimal.
Hence, $10\pi =\dfrac{10\times 22}{7}=31.428$
Now the number before the decimal will be degree which is 31 and the for the number after decimal we will convert it to minute using (1).
Now we have $0.428{}^\circ$ and using (1) we get,
$0.428{}^\circ =60\times 0.428\min =25.68\min$
The number before the decimal will be a minute value.
Now from this we have 25min and we will convert 0.68 min to second.
We know that 1 sec = $\dfrac{1}{60}$ of arcminute.
Therefore we get,
$0.68\min =0.68\times 60\sec =40.8\sec$
The number before decimal will be the second value.
Hence, from this we get 40sec.
Therefore, the answer in degree, minutes and seconds is: $31{}^\circ 25'40''$ .

Note: Here the conversion method that we have used to convert degrees to minutes and seconds is very important. The end result that we have got must be remembered to avoid any mistakes while solving the question.