Question

Find the radian measure of \[{{175}^{\circ }}4{5}'\]
(a) \[\dfrac{700}{720}\pi \]
(b) \[\dfrac{703}{720}\pi \]
(c) \[\dfrac{705}{720}\pi \]
(d) \[\dfrac{710}{720}\pi \]

Answer 1

Hint: We solve this problem by using the simple conversions of angles. First we use the conversion of minutes to degree in degree system that is
\[{{1}^{\circ }}=6{0}'\]
By using this result we convert the given angle into degree system then we use the other conversion that is
\[\dfrac{\pi }{2}={{90}^{\circ }}\]
By using this result we convert the angle from degree system to radian system.

Complete step-by-step answer:
We are given with the angle that is \[{{175}^{\circ }}4{5}'\]
Let us assume that the measure of given angle as\['A'\] that is
\[\Rightarrow A={{175}^{\circ }}4{5}'\]
We know that the conversion of minutes to degrees in degree system that is
\[\begin{align}
  & \Rightarrow {{1}^{\circ }}=6{0}' \\
 & \Rightarrow {1}'={{\left( \dfrac{1}{60} \right)}^{\circ }} \\
\end{align}\]

By using the above conversion to given angle we get
\[\begin{align}
  & \Rightarrow A={{175}^{\circ }}+45{{\left( \dfrac{1}{60} \right)}^{\circ }} \\
 & \Rightarrow A={{175}^{\circ }}+{{\left( \dfrac{3}{4} \right)}^{\circ }} \\
\end{align}\]
Now, by using the LCM method and adding the above numbers we get
\[\begin{align}
  & \Rightarrow A={{\left( \dfrac{4\times 175+3}{4} \right)}^{\circ }} \\
 & \Rightarrow A={{\left( \dfrac{703}{4} \right)}^{\circ }} \\
\end{align}\]
Now, we know that the conversion of degree system to radian system as
\[\begin{align}
  & \Rightarrow \dfrac{\pi }{2}={{90}^{\circ }} \\
 & \Rightarrow {{1}^{\circ }}=\dfrac{\pi }{180} \\
\end{align}\]
By using this conversion to above equation we get
\[\begin{align}
  & \Rightarrow A=\dfrac{703}{4}\times \dfrac{\pi }{180} \\
 & \Rightarrow A=\dfrac{703}{720}\pi \\
\end{align}\]
Therefore the radian measure of given angle is
\[\therefore {{175}^{\circ }}4{5}'=\dfrac{703}{720}\pi \]
So, option (b) is the correct answer.

So, the correct answer is “Option (b)”.

Note: Students may do mistake in converting the minutes angle to degrees.
Here we have the value of angle as
\[\Rightarrow A={{175}^{\circ }}4{5}'\]
We know that the conversion of minutes to degrees in degree system that is
\[\begin{align}
  & \Rightarrow {{1}^{\circ }}=6{0}' \\
 & \Rightarrow {1}'={{\left( \dfrac{1}{60} \right)}^{\circ }} \\
\end{align}\]
By using the above conversion to given angle we get
\[\begin{align}
  & \Rightarrow A={{175}^{\circ }}+45{{\left( \dfrac{1}{60} \right)}^{\circ }} \\
 & \Rightarrow A={{175}^{\circ }}+{{\left( \dfrac{3}{4} \right)}^{\circ }} \\
\end{align}\]
This is the correct procedure.
But students may do mistake after converting the minutes to degrees and take sthe value of the angle as
\[\begin{align}
  & \Rightarrow A={{175}^{\circ }}\times 45{{\left( \dfrac{1}{60} \right)}^{\circ }} \\
 & \Rightarrow A={{175}^{\circ }}\times {{\left( \dfrac{3}{4} \right)}^{\circ }} \\
\end{align}\]
This is because the representation of angle \[{{175}^{\circ }}4{5}'\] makes to think that degrees and minutes are in multiplication form. But this is wrong.
The representation of angle \[{{175}^{\circ }}4{5}'\] says that the degrees and minutes are in mixed fraction.
This point need to be taken care.