Question

Convert $3$ radian into degree.

Answer 1

Hint: Convert the units of radian into degree by using the formula $1\;{\text{rad}} = \dfrac{{180^\circ }}{\pi }$. Use the concepts of circle as the full angle is $2\pi {\text{ rad}}$ and this full angle in degrees is $360^\circ $.

Complete step-by-step answer:
The radian is used to measure angles used mainly in trigonometry. It is used instead of degrees. A Full circle has $2\pi \;{\text{rad}}$. A radian is equal to an angle to capture an arc of the circumference of the circle whose radius is $360^\circ $. One Radian is the angle subtended by an arc at the centre of a circle. Radians and degrees are both used in measuring angles. One-pie radian represents going $180^\circ $ around a circle which makes $\dfrac{{180}}{\pi }$.
The angle is made by taking the radius and covering it round the circle. Radians are the empathetic way to find the angles. One radian is equal to the angle created by taking the radius and stretching it along the edge of the circle. Instead of degrees, radian gives better results.
It is known that,
$1\;{\text{rad}} = \dfrac{{180^\circ }}{\pi }$
Therefore, $3$ radian is converted into degree as,
$\Rightarrow$$3\;{\text{rad}} = \dfrac{{180 \times 3}}{\pi }$
Now we substitute the value of $\pi $ that is $\dfrac{{22}}{7}$,
$\Rightarrow$$3\;{\text{rad}} = \dfrac{{540}}{{22}} \times 7$
$\Rightarrow$$3\;{\text{rad}} = 171\dfrac{9}{{11}}$
It can be written as,
$\Rightarrow$$3\;{\text{rad}} = 171^\circ + \dfrac{{9^\circ }}{{11}}$
We know that,
$1^\circ = 60\;{\text{minutes}}$ and it is denoted by (‘).
Therefore,
$\Rightarrow$${171^\circ} + \dfrac{{9^\circ }}{{11}} = {171^\circ} + {\dfrac{{9 \times 60}}{{11}}'}$
$\Rightarrow$$171^\circ + \dfrac{{9^\circ }}{{11}} = 171^\circ + \dfrac{{{{540}'}}}{{11}}$
Further simplify as,
$\Rightarrow$\[171^\circ + {49'} + \dfrac{{{1'}}}{{11}} = 171^\circ + {49'} + \dfrac{{1 \times {{60}{''}}}}{{11}}\]
$\Rightarrow$\[171^\circ + {49'} + \dfrac{{{1'}}}{{11}} = 171^\circ + {49'} + {6{''}}\]

Hence, the $3\;{\text{rad}}$ in degree is $171^\circ {49'}{6{''}}$.

Note: Carefully apply the formula to convert whether Radian into degree or degree into radian and simplify it by converting it into minutes and seconds. The concept of radian and degree is used in many fields such as, physics, by astronomers, by scientists, etc. Degree measures angles by how far we titled our heads. Radian measures angle by distance travelled.