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:
 $$\theta ={\displaystyle \frac{r}{s}}$$
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 :
$${\pi }^c$$ $$=$$ $${180}^{\circ }$$
$$\Rightarrow$$ $$1^c$$ $$=$$ $${\left({\displaystyle \frac{180}{\pi }}\right)}^{\circ }$$
Substitute the value of $$\pi$$ in the above equation:
$$\Rightarrow$$ $$1^c$$ $$=$$ $${\left({\displaystyle \frac{180}{\left({\displaystyle \frac{22}{7}}\right)}}\right)}^{\circ }$$
$$\Rightarrow$$ $$1^c$$ $$=$$ $${\left({\displaystyle \frac{180\times 7}{22}}\right)}^{\circ }$$
$$\Rightarrow$$ $$1^c$$ $$\approx$$ $$57.2958$$
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.