Express in terms of a right angle, the angle \[{{75}^{\circ }}15'\].

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:
\[1'={{\left( \dfrac{1}{60} \right)}^{\circ }}\]
After converting the given angle completely into degree 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’.

Complete step-by-step answer:
We have been given the angle \[{{75}^{\circ }}15'\] which we have to express in terms of right angle.
We know that \[1'={{\left( \dfrac{1}{60} \right)}^{\circ }}\]
\[\Rightarrow 15'={{\left( 15\times \dfrac{1}{60} \right)}^{\circ }}={{0.25}^{\circ }}\]
So \[{{75}^{\circ }}15'\] is equal to \[\left( {{75}^{\circ }}+15' \right)\].
\[\Rightarrow {{75}^{\circ }}15'=\left( {{75}^{\circ }}+15' \right)\]
Substituting the value of 15’ in the terms of degree, we get as follows:
\[\Rightarrow {{75}^{\circ }}15'=\left( {{75}^{\circ }}+{{0.25}^{\circ }} \right)={{75.25}^{\circ }}\]
Let us suppose \[{{75.25}^{\circ }}\] is equal to k times a right angle \[\left( {{90}^{\circ }} \right)\].
\[\Rightarrow {{75.25}^{\circ }}=k\times {{90}^{\circ }}\]
On dividing the equation by \[{{90}^{\circ }}\], we get as follows:
\[\begin{align}
  & \Rightarrow \dfrac{{{75.25}^{\circ }}}{{{90}^{\circ }}}=\dfrac{k\times {{90}^{\circ }}}{{{90}^{\circ }}} \\
 & \Rightarrow \dfrac{301}{360}=k \\
\end{align}\]
Therefore, the angle \[{{75}^{\circ }}15'\] can be expressed in terms of a right angle as \[\dfrac{301}{360}\] 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. Then suppose the angle to be equal to k times the 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 is equal to \[\dfrac{\pi }{2}\] radians. Also, sometimes in order to calculate the values of ‘k’, we just divide \[{{90}^{\circ }}\] the given angle by mistake. So be careful while calculating the value of k.