Question

Express in terms of a right angle, the angle \[{{130}^{\circ }}30'\].

Answer 1

Hint: We know that a right angle is equal to \[{{90}^{\circ }}\]. We will first convert the given angle in degrees completely for which we will use the following conversion:
\[1'={{\left( \dfrac{1}{60} \right)}^{\circ }}\]

Complete step-by-step answer:
After converting the given angle completely into degrees, we will suppose the given angle is equal to ‘k’ times the right angle and thus equating them we will find the value of ‘k’.
We have been given the angle \[{{130}^{\circ }}30'\] which we have to express in terms of right angle.
We know that \[1'={{\left( \dfrac{1}{60} \right)}^{\circ }}\]
\[\Rightarrow 30'={{\left( 30\times \dfrac{1}{60} \right)}^{\circ }}={{\left( \dfrac{1}{2} \right)}^{\circ }}\]
\[\Rightarrow {{130}^{\circ }}30'={{130}^{\circ }}+30'\]
Substituting the value of 30’ in degrees, we get as follows:
\[\Rightarrow {{130}^{\circ }}30'={{130}^{\circ }}+{{0.5}^{\circ }}={{130.5}^{\circ }}\]
Let us suppose \[{{130.5}^{\circ }}\] is equal to k times a right angle.
\[\Rightarrow {{130.5}^{\circ }}=k\times {{90}^{\circ }}\]
On dividing the above equation by \[{{90}^{\circ }}\], we get as follows:
\[\begin{align}
  & \Rightarrow \dfrac{{{130.5}^{\circ }}}{{{90}^{\circ }}}=\dfrac{k\times {{90}^{\circ }}}{{{90}^{\circ }}} \\
 & \Rightarrow \dfrac{29}{20}=k \\
\end{align}\]
Therefore, the angle \[{{130.5}^{\circ }}\] can be expressed in terms of a right angle as \[\dfrac{29}{20}\] times of a right angle.

Note: In this type of questions, first of all check that the given angle is completely either in degree or radians as the unit of measurement must be the same. One of the most common mistake that one might make while converting the minute to degree is by writing $1'={{60}^{\circ }}$, but it is wrong and actually, ${{1}^{\circ }}=60'$ hence, \[1'={{\left( \dfrac{1}{60} \right)}^{\circ }}\].Also, sometimes in order to calculate the value of ‘k’ we just divide by \[{{90}^{\circ }}\] by the given angle by mistake. So, be careful while calculating the value of k.