Question

One radian is equal to following degree measure
(A) \[{{57}^{o}}{{14}^{'}}{{21}^{''}}\]
(B) \[{{57}^{o}}{{16}^{'}}{{22}^{''}}\]
(C) \[{{58}^{o}}{{14}^{'}}{{21}^{''}}\]
(D) \[{{58}^{o}}{{16}^{'}}{{22}^{''}}\]

Answer 1

Hint: We are given a question based on unit conversion. We are given one radian and we are asked to find a degree measure equivalent of the same from the options given. We know that in order to convert radians to degree measure, we have to use the formula, \[\text{Degree measure}=\dfrac{180}{\pi }\times \text{Radian measure}\]. Here, we will take the value of \[\pi =\dfrac{22}{7}\] . We will then substitute the values and get the degree measure value in degrees. We will use \[{{1}^{\circ }}={{60}^{'}}\] and \[{{1}^{\circ }}={{3600}^{''}}\]to further the value in minutes and seconds. Hence, we will have the required value.

Complete step by step answer:
According to the given question, we are given one radian and we are asked to convert it to a degree measure and choose the most appropriate option from the given.
\[1radian\]---(1)
To make the conversion from the radian measure to degree measure, we will make use of formula, which is,
\[\text{Degree measure}=\dfrac{180}{\pi }\times \text{Radian measure}\]
 Now, substituting the values in the above formula, we get,
\[\Rightarrow \dfrac{180}{\pi }\times 1\]
Here, we will take the value of \[\pi =\dfrac{22}{7}\], we get,
\[\Rightarrow \dfrac{180}{\left( \dfrac{22}{7} \right)}\]
\[\Rightarrow \dfrac{180\times 7}{22}\]
We will have to simplify it, we have,
\[\Rightarrow \dfrac{90\times 7}{11}=\dfrac{630}{11}\]
\[\Rightarrow {{57.27}^{\circ }}\]
We got the answer degrees, we want the answer in minutes and seconds as well. We know that, \[{{1}^{\circ }}={{60}^{'}}\] and \[{{1}^{\circ }}={{3600}^{''}}\], we will use these and so we get,
\[\Rightarrow {{57}^{\circ }}+{{0.27}^{\circ }}\]
We will multiply by 60 to 0.27 degrees to get the answer in minutes and so we have,
\[\Rightarrow {{57}^{\circ }}+0.27\times 60\]
\[\Rightarrow {{57}^{\circ }}+{{16.2}^{'}}\]
Similarly, we will multiply 60 to the 0.2 minutes to get the answer in seconds, we get,
\[\Rightarrow {{57}^{\circ }}+{{16}^{'}}+{{0.2}^{'}}\]
\[\Rightarrow {{57}^{\circ }}+{{16}^{'}}+0.2\times 60\]
\[\Rightarrow {{57}^{\circ }}+{{16}^{'}}+{{12}^{''}}\]
\[\Rightarrow {{57}^{\circ }}{{16}^{'}}{{12}^{''}}\]

So, the correct answer is “Option B”.

Note: The conversion from degrees measure to radian measure can be done by using the formula, \[\text{Radian measure}=\dfrac{\pi }{180}\times \text{Degree measure}\]. The calculations must be done step wise. Also, the decimal part of the degree measure was used to express in minutes and the decimal part of the later was used to express in seconds. The answer that we obtained was close enough or approximately equal only to the option (A), hence, we have our answer.