Question

How do you convert 3 rad to degrees measurement?

Answer 1

Hint:
To convert radians to degrees use the formula $1\,rad = \dfrac{{{{180}^\circ }}}{\pi }$, where \[^\circ \] denotes degrees and also remember that a full circle that is $2\pi $ radians which is equal to 360 degrees.

Complete step by step solution:
The objective of the problem is to convert radians into degrees measurements.
About radians : A radian is defined as the angle subtended by an arc of length equal to the radius of the circle at its centre. A radian is a unit of angle measurement. The radian measure is also called a circular measure , as it is related to a circle.
About degrees : If a rotation from the initial side to the terminal side is 1/360 th of one revolution, the angle is said to have a measure of one degree. It is written as ${1^\circ }$. A degree is further divided into sixty equal parts and each part is called one minute, denoted as $1'$ . and each minute is further divided into 60 equal parts and each part is called as one second. It is written as $1''$.
Now we are going to convert 3 radians into degrees.
We know that $1\,rad = \dfrac{{{{180}^\circ }}}{\pi }$
To convert 3 radians to degrees measure we should multiply 3 radians with the one radian measure .
One multiplying we get
\[
  3 \times 1\,rad = 3 \times \dfrac{{{{180}^\circ }}}{\pi } \\
   \Rightarrow 3\,rad = \dfrac{{{{540}^\circ }}}{\pi } \\
 \]
We know that the value of pi is 22/7, substitute this value in above equation we get
$
   \Rightarrow 3\,rad = \dfrac{{{{540}^\circ }}}{{\dfrac{{22}}{7}}} \\
   \Rightarrow 3\,rad = \dfrac{{{{540}^\circ } \times 7}}{{22}} \\
 $
On simplifying we get, $3\,rad = 171\dfrac{{{9^\circ }}}{{11}}$
As mentioned above we can take ${1^\circ } = 60'\,,\,\,1' = 60''$
We can write $3\,rad = 171\dfrac{{{9^\circ }}}{{11}}$ as $3\,rad = 171 + \dfrac{{{9^\circ }}}{{11}}$
Substituting ${1^\circ } = 60'\,,\,\,1' = 60''$ in above equation . we get
$
  3\,rad = 171 + \dfrac{{9 \times {{60}^{'}}}}{{11}} \\
  \,\,\,\,\,\,\,\,\,\,\,\, = 171 + \dfrac{{{{540}^{'}}}}{{11}} \\
 $
540/11 can be written in mixed fraction as $49\dfrac{{1'}}{{11}}$. This can be written as $49' + \dfrac{{1'}}{{11}}$
Now we get
  $
  3\,rad = {171^\circ } + 49' + \dfrac{{1'}}{{11}} \\
  3\,rad = {171^\circ } + 49' + \dfrac{{1 \times 60''}}{{11}} \\
  3rad = {171^\circ } + 49' + 6'' \\
 $

Thus, 3 radians is equal to ${171^\circ }49'6''\,$ degrees.

Note:
It is important to remember that the radian is not dependent on the radius of the circle. A radian is a measure of an angle. Measure of an angle is a real number. The angle which intercepts an arc of length one unit of a unit circle is called one radian.