Question

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

Answer 1

Hint: 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 \[1\].