Question

How to convert $(-3)radian$ to Degrees-minutes-second?

Answer 1

Hint: We are asked to convert $(-3)radian$ to Degrees-minutes-second. We know that, \[2\pi \] radians \[={{360}^{\circ }}\]. From this expression we can find the value of 1 radian which is, \[\dfrac{{{360}^{\circ }}}{2\pi }\]. Now, we will multiply the value of 1 radian by (-3) and then we will solve it further to find the value in degree-minutes-seconds.

Complete step-by-step answer:
According to the given question, we are given a value in radians and we are to express it in degrees-minutes-seconds.
Radian is a unit of angle and is defined as the angle subtended at the centre of a circle by the arc of the circle.
Degree is another unit of angle. So, basically we have to express one unit in terms of the other.
We have to convert $(-3)radian$ to Degrees-minutes-second.
We know that,
\[2\pi rad.={{360}^{\circ }}\]
So, \[1rad.=\dfrac{{{360}^{\circ }}}{2\pi }\]
\[\Rightarrow 1rad.=\dfrac{{{180}^{\circ }}}{\pi }\]
We have the value of 1 radian, so to get the value of (-3) radians we will multiply the value of 1 radian by (-3). But carrying around negative sign will be tedious. So, we will now multiply by 3. And afterwards we will apply the negative sign.
\[3\pi rad.=\dfrac{{{180}^{\circ }}}{\pi }\times 3\]
\[\Rightarrow \dfrac{180}{\pi }\times 3=\dfrac{180\times 3}{\pi }\]
\[=\dfrac{240}{\pi }\]
We know that, commonly used value of \[\pi =\dfrac{22}{7}\] or \[\pi =\dfrac{355}{113}\], in this question we will use \[\pi =\dfrac{355}{113}\].
We get,
\[=\dfrac{240\times 113}{355}\]
\[=\dfrac{27120}{355}\]
Dividing the numerator and the denominator by 5, we get,
\[=\dfrac{5424}{71}\]
Writing in terms of mixed fraction, we get,
\[=76{{\dfrac{28}{71}}^{\circ }}\]
Or
\[={{76}^{\circ }}+{{\dfrac{28}{71}}^{\circ }}\]
We have the angle in degrees, and require the fraction to be in minutes. For that, we will \[{{\dfrac{28}{71}}^{\circ }}\] by 60, we get,
\[\dfrac{28}{71}\times 60\]
\[=\dfrac{28\times 60}{71}\]
\[=\dfrac{1680}{71}\]
Writing the above expression in mixed fraction, we get,
\[=23{{\dfrac{47}{71}}^{'}}\]
Or
\[=23'+{{\dfrac{47}{71}}^{'}}\]
We have the angle in minutes, now we have to convert the rest of the fraction in seconds. And for that, we will be multiplying \[{{\dfrac{47}{71}}^{'}}\] by 60, we get,
\[\dfrac{47}{71}\times 60\]
\[=\dfrac{47\times 60}{71}\]
\[=\dfrac{2820}{71}\]
Writing as a mixed fraction, we get,
\[=39{{\dfrac{51}{71}}^{''}}\]
Or
\[=39''+{{\dfrac{51}{71}}^{''}}\]
Therefore, \[(-3)radians=-{{76}^{\circ }}23'39''\]

Note: The calculation for the above question is lengthy, carry out the solution step wise to prevent any errors. We deliberately did not apply a negative sign in the beginning as it would have made the calculation complex. So, we omitted it during our calculation and applied it only at the end.