Question

How does one convert $15$ degree $30'$ into radians?

Answer 1

Hint: Firstly, convert minutes into degrees using ${1^ \circ } = 60'$ then add it and then use the relationship we have between degrees and radians i.e., ${1^ \circ } = \dfrac{\pi }{{180}}$ radians, with this we can easily convert our degrees expression into radians.

Formula Used:
We used ${1^ \circ } = \dfrac{\pi }{{180}}$ radians,
We also used ${1^ \circ } = 60'$ or one degree is equal to sixty minutes.

Complete step by step solution:
We know that ${1^ \circ } = \dfrac{\pi }{{180}}$ radians,
And we have to convert ${15^ \circ }30'$ , for this we first convert this in degree only,
We also know that ${1^ \circ } = 60'$
Or $1' = {\dfrac{1}{{60}}^ \circ }$
Therefore, we can say that
$30' = {\dfrac{{30}}{{60}}^ \circ } \\
\Rightarrow 30' = {0.5^ \circ } \\ $
Now we have ${15^ \circ } + {0.5^ \circ }$ i.e., ${15.5^ \circ }$
As we already know ,
${1^ \circ } = \dfrac{\pi }{{180}}rad \\
\Rightarrow {15.5^ \circ } = 15.5 \times \dfrac{\pi }{{180}}rad \\
\Rightarrow 0.08611\pi \, rad \\
\therefore 0.27038\,rad $
And here we get our answer by simply applying the formula , similarly we can convert many more conversions .

Additional Information: We can represent one full revolution by $2\pi $ in radians and ${360^ \circ }$ in degrees.
$\Rightarrow 2\pi = {360^ \circ } \\
\Rightarrow \pi = {180^ \circ }$
We measure angles in degrees in mathematics of geometry and we use radians commonly in trigonometric function or periodic function. A degree is further divided into other parts , namely minutes and seconds. And they have the following relationship,
${1^ \circ } = 60'and \\
1' = 60'' $
Or one degree is equal to sixty minutes and one minute is equal to sixty seconds. Or we can say that one degree is equivalent to sixty minutes or three hundred sixty seconds.

Note: We always represent radians in terms of $\pi $ (pi), And this $\pi = \dfrac{{22}}{7} = 3.14$ we use this as per our convenience . One degree is equal to \[0.0174533\] radians or we can say that one radian is equal to \[57.2958\] degrees. We can convert one to another by using this simple formula .