Question

How do you convert \[{72^0}\] to radians?

Answer 1

Hint: The measurement of angles can be done in two different units namely radian and degree. In geometry, we measure the angles in degree but also in radians sometimes, similarly in trigonometry, we measure the angle in radians but sometimes in degrees too. So, there are different kinds of units for determining the angle that are, degrees and radians. There is a simple formula to convert a given degree into radians (vice versa). Using that formula, we can find out the correct answer.

Complete step by step solution:
The value of \[{180^0}\] is equal to \[\pi \]radians.
Then \[{1^0}\] is equal to \[\dfrac{\pi }{{180}}\] radians.
So the given \[{x^0}\] is equal to \[x \times \dfrac{\pi }{{180}}\] radians.
This is the general formula for converting the angel in degrees to radians.
Now We have, \[{72^0}\].
Using the formula
\[{72^0} = 72 \times \dfrac{\pi }{{180}}{\text{ radians}}\]
\[ = \dfrac{{72\pi }}{{180}}\].
To cancel this we find the factors of 72 and 180.
That is, \[72 = 2 \times 2 \times 2 \times 9\]
\[180 = 2 \times 2 \times 5 \times 9\]
Then we have,
\[ = \dfrac{{2 \times 2 \times 2 \times 9}}{{2 \times 2 \times 5 \times 9}}\pi \]
Cancelling we have
\[ = \dfrac{{2\pi }}{5}\].
Hence \[{72^0}\] is \[\dfrac{{2\pi }}{5}\] rad.
We can put it in the decimal form, that is we know that the value of \[\pi \] is 3.142.
Substituting and simplifying we have,
\[ = \dfrac{{2 \times 3.142}}{5}\]
\[ = \dfrac{{6.284}}{5}\]
\[ = 1.2568\].
Hence \[{72^0}\] is 1.2568 rad.

Note: We know that the radian is denoted by ‘rad’. Suppose if they ask us to convert \[\dfrac{{2\pi }}{5}\]rad into degrees. The value of \[\pi \] radian is equal to \[{180^0}\]. Then 1 rad is equal to \[\dfrac{{180}}{\pi }\] degrees. So the given \[x\] rad is equal to \[x \times \dfrac{{180}}{\pi }\] degrees. This is the general formula for converting the angle in radians to degrees. Then \[\dfrac{{2\pi }}{5}\] rad becomes
\[\dfrac{{2\pi }}{5} = \dfrac{{2\pi }}{5} \times \dfrac{{180}}{\pi }\] degree
\[ = \dfrac{{2 \times 180}}{5}\]
\[ = 2 \times 36\]
\[ = {72^0}\].
Hence \[\dfrac{{2\pi }}{5}\]rad is equal to \[{72^0}\]