Question

The degree measure of 1 radian (taking $\pi = \dfrac{{22}}{7}$) is-
$
  A{\text{ 5}}{{\text{5}}^ \circ }61'22''\left( {approx.} \right) \\
  B{\text{ 5}}{{\text{7}}^ \circ }16'22''\left( {approx.} \right) \\
  C{\text{ 5}}{{\text{7}}^ \circ }22'16''\left( {approx.} \right) \\
  D{\text{ 5}}{{\text{7}}^ \circ }22'22''\left( {approx.} \right) \\
 $

Answer 1

Hint- Here we will proceed by using $\pi {\text{ radians = 18}}{0^ \circ }$ to find integer part where integer part is the degree. Then we will multiply the integer part with 60’ to convert into minutes and further convert minutes into seconds by multiplying with 60’’. Hence we will get the desired result.

Complete step-by-step solution -
One radian is the measure of the central angle whose arc length is the same as the radius of the circle.
As we know that $\pi {\text{ radians = 18}}{0^ \circ }$
And 1 radian $ = \dfrac{{180}}{\pi } = \dfrac{{180}}{{\dfrac{{22}}{7}}}$
$ \Rightarrow $1 radian = 57.272727
Here the integer part i.e. 57.272727 constitutes the degree part.
And we will convert mantissa into minutes by multiplying with 60’.
Therefore minutes $ = 0.272727 \times 60' = 16.3636'$
Now the integer part i.e. 0.272727 constitutes the minutes.
And we will convert mantissa into seconds by multiplying with 60’’.
Therefore seconds $ = 0.3636 \times 60'' = 22''$
Hence the degree measure of 1 radian is ${57^ \circ }16'22''$.
$\therefore $ Option B is correct.

Note- While solving this question, many of us omit the step of converting integer part into minutes. This will lead to inappropriate results. Therefore, we must know all the steps of conversion of 1 radian into degree. Hence we will get the required result.