Question

How do you convert 3 rad to degrees measurement?

Answer 1

Hint:
To convert radians to degrees use the formula $1rad={\displaystyle \frac{{180}^{\circ }}{\pi }}$, 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 $1^{\prime }$ . 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 $1rad={\displaystyle \frac{{180}^{\circ }}{\pi }}$
To convert 3 radians to degrees measure we should multiply 3 radians with the one radian measure .
One multiplying we get
$$\begin{gathered} 3\times 1rad=3\times {\displaystyle \frac{{180}^{\circ }}{\pi }} \\ \Rightarrow 3rad={\displaystyle \frac{{540}^{\circ }}{\pi }} \end{gathered}$$
We know that the value of pi is 22/7, substitute this value in above equation we get
$$\begin{gathered} \Rightarrow 3rad={\displaystyle \frac{{540}^{\circ }}{{\displaystyle \frac{22}{7}}}} \\ \Rightarrow 3rad={\displaystyle \frac{{540}^{\circ }\times 7}{22}} \end{gathered}$$
On simplifying we get, $3rad=171{\displaystyle \frac{9^{\circ }}{11}}$
As mentioned above we can take $1^{\circ }={60}^{\prime },1^{\prime }={60}^{″}$
We can write $3rad=171{\displaystyle \frac{9^{\circ }}{11}}$ as $3rad=171+{\displaystyle \frac{9^{\circ }}{11}}$
Substituting $1^{\circ }={60}^{\prime },1^{\prime }={60}^{″}$ in above equation . we get
$$\begin{gathered} 3rad=171+{\displaystyle \frac{9\times {60}^{{}^{\prime }}}{11}} \\ =171+{\displaystyle \frac{{540}^{{}^{\prime }}}{11}} \end{gathered}$$
540/11 can be written in mixed fraction as $49{\displaystyle \frac{1^{\prime }}{11}}$. This can be written as ${49}^{\prime }+{\displaystyle \frac{1^{\prime }}{11}}$
Now we get
  $$\begin{gathered} 3rad={171}^{\circ }+{49}^{\prime }+{\displaystyle \frac{1^{\prime }}{11}} \\ 3rad={171}^{\circ }+{49}^{\prime }+{\displaystyle \frac{1\times {60}^{″}}{11}} \\ 3rad={171}^{\circ }+{49}^{\prime }+6^{″} \end{gathered}$$

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