Question

Convert $6$ radians into degree measure.

Answer 1

Hint: This is a problem which can be solved by a unitary method. we know a very common relation between degree and radian that is ${180^ \circ } = \pi $ radians. From this relation we have to first calculate the value of $1$ radian into degree then multiply the obtained value by $6$ to get the value of $6$ radian into degree.

Complete step-by-step solution:
Here, we have to convert $6$ radians into degrees.
We know a very common relation between degree and radian that is ${180^ \circ } = \pi $ radians.
Now, by applying the procedure of unitary methods.
$\because \pi $ radians are equal to ${180^ \circ }$.
$\therefore 1$ radian is equal to $\dfrac{{{{180}^ \circ }}}{\pi }$ degree.
$\therefore 6$ radians are equal to $\dfrac{{{{180}^ \circ }}}{\pi } \times 6$ degree.
Now, substitute the value of $\pi = \dfrac{{22}}{7}$ in the above expression to get the required value.
$\therefore 6$ radians are equal to $\dfrac{{{{180}^ \circ }}}{{\dfrac{{22}}{7}}} \times 6 = \dfrac{{{{180}^ \circ } \times 7}}{{22}} \times 6 = \dfrac{{{{3780}^ \circ }}}{{11}} = 343.6$ degree.
Thus, the value of $6$ radians is $343.6$ degree.
This result can also be put forward into degree, minutes and second.
We get the degree measure for $6$radians as $ = \dfrac{{{{180}^ \circ } \times 6 \times 7}}{{22}} = \dfrac{{{{3780}^ \circ }}}{{11}}$
Now, by dividing $3780$ by $11$. we get,
\[
  \,\,\,\,\,\,\underline {343} \\
  \left. {11} \right)3780 \\
  \,\,\,\,\,\,\,\underline {33} \\
  \,\,\,\,\,\,\,\,\,\,48 \\
  \,\,\,\,\,\,\,\,\,\,\underline {44} \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,40 \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,\underline {33} \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,7 \\
 \]
Degree measure $ = \left( {{{343}^ \circ } + \dfrac{{{7^ \circ }}}{{11}}} \right)$
We know that $1$ degree is equal to $60$ minutes. Now converting $\dfrac{7}{{11}}$ degree into minutes. we get,
Degree measure $ = 34{3^ \circ } + {\left( {\dfrac{7}{{11}} \times 60} \right)'} = 34{3^ \circ } + {\left( {\dfrac{{420}}{{11}}} \right)'}$
Now, divide $420$ by $11$
$
  \,\,\,\,\,\,\,\,38 \\
  \left. {11} \right)\overline {420} \\
  \,\,\,\,\,\,\,\underline {33} \\
  \,\,\,\,\,\,\,\,\,\,90 \\
  \,\,\,\,\,\,\,\,\,\,\underline {88} \\
  \,\,\,\,\,\,\,\,\,\,\,\,2 \\
 $
Degree measure $ = 34{3^\circ } + {38'} + {\left( {\dfrac{2}{{11}}} \right)'}$
We also know that $1$ minute is equal to $60$ seconds. Now converting $\dfrac{2}{{11}}$ minutes into second we get,
Degree measure $ = {343^ \circ } + {38'} + {\left( {\dfrac{2}{{11}} \times 60} \right){''}} = {343^ \circ } + {38'} + {\left( {10.9} \right){''}}$

Thus, $6$ radians are approximately equal to \[{343^ \circ } + {38'} + 1{1{''}}\]

Note: We measure angles in unit degree and radians.
In the problem of unitary method our main aim is to find the price of one object then to calculate the price of a number of given objects we have to simply multiply the price of one object and the given number of objects.