Question

Find radian measure corresponding to the degree measure ${{37}^{\circ }}3{0}'$.

Answer 1

Each degree is divided into ${{60}^{\circ }}$ equal minutes and each minute is further divided into equal $60\text{ }seconds$.
The relation between degree and radian is given by the formula, ${{1}^{\circ }}=\dfrac{\pi }{180}\text{radians}$ where $\pi$ is a constant whose value is equal to approximately $3.14$.

Complete step-by-step answer:
The degree given in the question is ${{37}^{\circ }}3{0}'$.
Now as we all know that the angle, ${{1}^{\circ }}$ when converted to minutes the value is $60'$, hence, using the conversion value of the ${{1}^{\circ }}$, we find the value of $3{0}'$ given us as:
$\Rightarrow 3{0}'={{\left( \dfrac{1}{60}\times 30 \right)}^{\circ }}$
$={{0.5}^{\circ }}$
Now, adding the minutes converted degree to the degree value of ${{37}^{\circ }}$. The total value of the degree measures as:
${{37}^{\circ }}30'={{37}^{\circ }}+{{0.5}^{\circ }}$
$={{37}^{\circ }}{{.5}^{\circ }}$
Now we convert the degree value into radian to find the value in radian for a single degree first and then we will convert the rest accordingly. Hence, the value of ${{1}^{\circ }}$ is given as:
${{1}^{\circ }}=\dfrac{\pi }{180}\text{radians}$.
So, the radian measure corresponding to the degree measure ${{37.5}^{\circ }}$ after converting them into radian by multiplying them with $\dfrac{\pi }{180}$ we get the value as:
${{37.5}^{\circ }}=\left( \dfrac{\pi }{180}\times 37.5 \right)\text{radians}$
$=\text{0}\text{.6542}\ \text{radians}$

Hence, the radian value of the degree ${{37}^{\circ }}3{0}'$ is $\text{0}\text{.6542}\ \text{radians}$.

Note: Students may go wrong while converting the value from degree to radian, is that they might think that both $\pi$ and ${{180}^{\circ }}$ are same in this instance as although we use both for same purpose as in angular form $\pi$ is considered as ${{180}^{\circ }}$ but not here, here we need the value of $\pi$ which is $3.1415$ so they won’t cut themselves to reduced value of $1$.