Question

Express in terms of a right angle, the angle \[{{210}^{\circ }}30'30''\].

Answer 1

Hint: We know that the right angle is equal to \[{{90}^{\circ }}\]. First of all we will convert the given angle completely into degrees which means that we have to convert minutes and seconds into degrees for which we have the following conversions:
\[\begin{align}
  & 1'={{\left( \dfrac{1}{60} \right)}^{\circ }} \\
 & 1''={{\left( \dfrac{1}{3600} \right)}^{\circ }} \\
\end{align}\]

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 \[{{210}^{\circ }}30'30''\] which we have to express in terms of right angle.
We know that \[1'={{\left( \dfrac{1}{60} \right)}^{\circ }}\]
\[\Rightarrow 30'={{\left( 30\times \dfrac{1}{60} \right)}^{\circ }}={{\left( \dfrac{1}{2} \right)}^{\circ }}\]
Also we know that \[1''={{\left( \dfrac{1}{3600} \right)}^{\circ }}\]
\[\Rightarrow 30''={{\left( 30\times \dfrac{1}{3600} \right)}^{\circ }}={{\left( \dfrac{1}{120} \right)}^{\circ }}\]
\[\Rightarrow {{210}^{\circ }}30'30''={{210}^{\circ }}+30'+30''\]
Substituting the values of 30’ and 30’’ in degrees, we get as follows:
\[\Rightarrow {{210}^{\circ }}30'30''={{210}^{\circ }}+{{\dfrac{1}{2}}^{\circ }}+{{\dfrac{1}{120}}^{\circ }}\]
On taking LCM of 2 and 120, we get as follows:
\[\Rightarrow {{210}^{\circ }}30'30''=\dfrac{{{210}^{\circ }}\times 120+60+1}{120}=\dfrac{25200+60+1}{120}={{\dfrac{25261}{120}}^{\circ }}\]
Let us suppose the angle \[{{\dfrac{25261}{120}}^{\circ }}\] is equal to k times a right angle.
\[\Rightarrow {{\dfrac{25261}{120}}^{\circ }}=k\times {{90}^{\circ }}\]
On dividing the equation by \[{{90}^{\circ }}\] on both the sides, we get as follows:
\[\begin{align}
  & \Rightarrow {{\dfrac{25261}{120\times {{90}^{\circ }}}}^{\circ }}=\dfrac{k\times {{90}^{\circ }}}{{{90}^{\circ }}} \\
 & \Rightarrow k=\dfrac{25261}{10800} \\
\end{align}\]
Therefore, the angle \[{{210}^{\circ }}30'30''\] can be expressed in terms of right angle as \[\dfrac{25261}{10800}\] times a right angle.

Note: In this type of questions, first of all check that the given angle is completely either in degree or radians then suppose the angle to be equal to k times right angle and the unit of measurement must be the same. If the given angle is in degrees then you will take right angle equal to \[{{90}^{\circ }}\] and if the given angle is in radians then you will take right angle equal to \[\dfrac{\pi }{2}\] radians. 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.