Question

Convert 10 radians into degree minutes and seconds.

Answer 1

Hint:We are given an angle of 10 radians and we are supposed to convert it into degrees so we make it one by one by using formulae of conversion i.e. for converting radians into degree we multiply the angle in radian with ${\displaystyle \frac{180}{\pi }}$ .

Complete solution step by step:
Firstly we understand what radian means
A radian is considered an angle with an arc that has the same length as a circle 's radius.
So the circumference of a circle with radius = 1 has a length of $2\pi$ which means that$\pi {\text{rad = 180}}^{\circ }$.
Using the definition of radian we try to translate it into mathematically and have
Circumference = $2\pi r$
Radius $r=1$
$$\begin{gathered} \pi \text{rad}={180}^{\circ } \\ \Rightarrow 1\text{rad}={\displaystyle \frac{180}{\pi }} \end{gathered}$$
So we try to use this result and apply it in our question
$10\text{rad}=10\times {\displaystyle \frac{180}{\pi }}=572.9577$
Now we have converted the radians into ${572.9577}^{\circ }$ degrees.
Now converting the degrees further into minutes and seconds we multiply the decimal value like this
$$\begin{gathered} {572.9577}^{\circ }={572}^{\circ }+{0.9577}^{\circ } \\ \Rightarrow 0.9577\times 60=57.462 \end{gathered}$$
We have got our result as $57.462$ minutes. Now we want to convert our minutes into seconds so we have
$$\begin{gathered} {57.462}^{\prime }={57}^{\prime }+{0.462}^{\prime } \\ \Rightarrow 0.462\times 60=27.72\approx {28}^{″} \end{gathered}$$
Finally we have got the seconds too i.e. ${28}^{″}$.
This means our complete angle in degree, minutes, second is
${572}^{\circ }{57}^{\prime }{28}^{″}$
Additional information: By the result we can say that the resultant angle in degrees represents more than one time around a circle because a complete round of circle means a distance of${360}^{\circ }$.

Note: We have used the value of Pi here in radians which is equal to ${180}^{\circ }$and this is why in trigonometry we use radian angles instead of degree angles because it is convenient to write.