Question

How many degrees are there in one radian?

Answer 1

Hint: This problem is related to unit conversion. We need to convert one radian to degrees. These are units of angle measurement. Use the relation \[{\pi ^c}\] \[ = \] \[{180^ \circ }\].

Complete step by step solution:
One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. That means, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle;
that is:
 \[\theta = \dfrac{r}{s}\]
Where \[\theta \] is the subtended angle in radians, \[s\] is arc length, and \[r\] is radius.
A degree is a measurement of a plane angle in which one full rotation is 360 degrees.
From the relation :
\[{\pi ^c}\] \[ = \] \[{180^ \circ }\]
\[ \Rightarrow \] \[{1^c}\] \[ = \] \[{\left( {\dfrac{{180}}{\pi }} \right)^ \circ }\]
Substitute the value of \[\pi \] in the above equation:
\[ \Rightarrow \] \[{1^c}\] \[ = \] \[{\left( {\dfrac{{180}}{{\left( {\dfrac{{22}}{7}} \right)}}} \right)^ \circ }\]
\[ \Rightarrow \] \[{1^c}\] \[ = \] \[{\left( {\dfrac{{180 \times 7}}{{22}}} \right)^ \circ }\]
\[ \Rightarrow \] \[{1^c}\] \[ \approx \] \[57.2958\]
The above value is often approximated as :
\[{1^c}\] \[ = \] \[{57^ \circ }\].

Note:
Both radian and degree are the units for measuring plane angle. A degree is symbolized by the ‘\[^ \circ \]’ symbol, while radian is symbolized by ‘\[^c\]’ symbol (both written in superscript). Radian is the S.I. unit for a plane angle.