Question

How do you convert ${{67.5}^{\circ }}$ into $\pi $ radians?

Answer 1

Hint: We are given a number as ${{67.5}^{\circ }}$ have to convert it into radian. To dos so, we will learn about the relation between the degree and radian, we will use the relation ${{180}^{\circ }}=\pi \text{radian}$ or we say $\pi \text{radian}={{180}^{\circ }}$ . To solve our problem, we will also use the unitary method to find the value of the given value.

Complete step by step solution:
We are given ${{67.5}^{\circ }}$degree and we are asked to convert it into radian.
Before we start solving, we will learn that dimensional analysis is a method which helps us in converting the term from one dimension to another dimension which actually changes the value of the quantity.
For example: we know that 1 meter is the same as 100cm, here quantity is same but dimension is different.
So we will use dimensional analysis,
We will look at the relation between degree and the radian. We get –
$180\text{degree}=\pi \text{radian}$
Or
We may write this as –
$\pi \text{radian}=180\text{degree}$
Now we will use a unitary method to find the value of 1 degree or 1 radian and then use if further.
Now, as we know that –
180 degrees is the same as $\pi $ radian.
$\Rightarrow {{180}^{\circ }}=\pi \text{radian}$
So, by unitary method, 1 degree will be given as –
$1\text{degree}=\dfrac{\pi }{{{180}^{\circ }}}\text{radian}$ ……………………………….(1)
Now we have to find the value of $67.5$ degree so since we have value of 1 degree as $\dfrac{\pi }{{{180}^{\circ }}}$ , so value of 67.5 degree will be achieved by multiplying $\dfrac{\pi }{{{180}^{\circ }}}$ by 67.5.
Hence, multiply equation (1) by 67.5. We get –
$67.5\text{degree}=\dfrac{\pi }{{{180}^{\circ }}}\times 67.5$
Now we simplify
${{67.5}^{\circ }}=\dfrac{\pi \times 67.5}{1800}$
Now we simplify $\dfrac{675}{1800}=\dfrac{135}{360}=\dfrac{27}{72}=\dfrac{3}{8}$ .
So, using this above we get –
${{67.5}^{\circ }}=\pi \times \dfrac{3}{8}\text{radian}$
So, we get –
The value of ${{37.5}^{\circ }}$ in $\pi $ radian as $\dfrac{3\pi }{8}$ radian.

Hence our answer is 67.5 degrees is the same as $\dfrac{3\pi }{8}$ radian.

Note: We need not to change $\pi $ into $\dfrac{22}{7}$ term, as mostly it will get cancelled in the process. Also we can always wait till the last moment. We do not direct start multiplying term is numerator or denominator because it will get long while we stay put and cancel as much as possible in numerator and denominator also remember unitary method is always help in find the value of ‘n’ item if we know value of item.