Question

Express in terms of right angles, and also in degrees, minutes and seconds, the angles:
(i) \[{{120}^{g}}\]
(ii) \[{{45}^{g}}35'24''\]

Answer 1

Hint: For the above question, we will have to know the conversion of grade, minutes and seconds into degrees as follows:
\[{{1}^{g}}\left( grade \right)={{\dfrac{9}{10}}^{\circ }}\]
\[1'\left( \operatorname{minute} \right)={{\dfrac{1}{60}}^{\circ }}\] or \[{{1}^{\circ }}=60\operatorname{minutes}\]
\[1''\left( \operatorname{seconds} \right)={{\dfrac{1}{3600}}^{\circ }}\] or \[{{1}^{\circ }}=3600\operatorname{seconds}\]

Complete step-by-step answer:
We also know that the right angle is equal to \[{{90}^{\circ }}\] . 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’.
The given angles are expressed in terms of right angles and also in degrees, minutes and seconds as follows:
(i) \[{{120}^{g}}\]
We have been given the angle 120 grade.
We know that \[1grade={{\left( \dfrac{9}{10} \right)}^{\circ }}\].
\[\Rightarrow 120grade={{\left( 120\times \dfrac{9}{10} \right)}^{\circ }}={{108}^{\circ }}\]
Let us suppose \[{{108}^{\circ }}\]to be equal to k times right angle.
\[\Rightarrow {{108}^{\circ }}=k\times {{90}^{\circ }}\]
On dividing the equation by \[{{90}^{\circ }}\] we get as follows:
\[\begin{align}
  & \Rightarrow \dfrac{108}{90}=k \\
 & \Rightarrow k=\dfrac{6}{5} \\
\end{align}\]
So \[{{108}^{\circ }}\] is equal to \[\dfrac{6}{5}\] times a right angle.
We also know that \[{{1}^{\circ }}=60'\].
\[\Rightarrow {{108}^{\circ }}=\left( 108\times 60 \right)'\operatorname{minutes}=6480minutes\]
Again, we know that \[{{1}^{\circ }}=3600\operatorname{seconds}\].
\[\Rightarrow {{108}^{\circ }}={{108}^{\circ }}\times 3600\operatorname{seconds}=388800seconds\]
Hence, \[{{120}^{g}}={{108}^{\circ }}=\dfrac{6}{5}\] of right angle = 6480 minutes = 388800 seconds
(ii) \[{{45}^{g}}35'24''\]
The angle is \[{{45}^{g}}35'24''\].
Since we know that \[1grade={{\left( \dfrac{9}{10} \right)}^{\circ }}\].
\[\Rightarrow 45grade=45\times \dfrac{9}{10}={{\dfrac{81}{2}}^{\circ }}\]
We also know that \[1'={{\left( \dfrac{1}{60} \right)}^{\circ }}\].
\[\Rightarrow 35'={{\left( 5\times \dfrac{1}{60} \right)}^{\circ }}={{\dfrac{7}{12}}^{\circ }}\]
We also know that \[1'={{\dfrac{1}{3600}}^{\circ }}\].
\[\Rightarrow 24'={{\left( 24\times \dfrac{1}{3600} \right)}^{\circ }}={{\left( \dfrac{1}{150} \right)}^{\circ }}\]
\[\Rightarrow {{45}^{g}}35'24''={{\left( \dfrac{81}{2} \right)}^{\circ }}+{{\left( \dfrac{7}{12} \right)}^{\circ }}+{{\left( \dfrac{1}{150} \right)}^{\circ }}\]
On taking LCM of 2, 12 and 150, we get as follows:
\[\Rightarrow {{45}^{g}}35'24''=\dfrac{81\times 150+7\times 25+2}{300}=\dfrac{12150+175+2}{300}=\dfrac{12327}{300}={{\dfrac{4109}{100}}^{\circ }}\]
Let us suppose \[{{\dfrac{4109}{100}}^{\circ }}\] is equal to k times the right angle.
\[\Rightarrow {{\dfrac{4109}{100}}^{\circ }}=k\times 90\]
On dividing the equation by \[{{90}^{\circ }}\] on both the sides, we get as follows:
\[\begin{align}
  & \Rightarrow \dfrac{4109}{100\times 90}=\dfrac{k\times 90}{90} \\
 & \Rightarrow k=\dfrac{4109}{9000} \\
\end{align}\]
We also know that \[{{1}^{\circ }}=60'\].
\[\Rightarrow {{\dfrac{4109}{100}}^{\circ }}=\left( \dfrac{4109}{100}\times 60 \right)\begin{matrix}
   ' \\
   {} \\
\end{matrix}=\dfrac{12327'}{5}(\operatorname{minutes})\]
Once again, we know that \[{{1}^{\circ }}=3600\operatorname{seconds}\].
\[\Rightarrow {{\dfrac{4109}{100}}^{\circ }}=\dfrac{4109}{100}\times 3600\operatorname{seconds}=147924seconds\]
Hence \[{{45}^{g}}35'24''=\dfrac{4109}{9000}\] times right angle \[=\dfrac{12327}{5}\operatorname{minutes}=147924seconds\].

Note: Sometimes we use \[{{1}^{\circ }}=\dfrac{9}{10}grade\] by mistake which is wrong. So, be careful while using it. Also, remember that the given angle must be completely into degree or radian then we will use the conversion according to it. If the given angle is in the form of \[{{x}^{\circ }}y'z''\] then first of all we will convert minutes and seconds into degrees. Also remember that a right angle is equal to 90 degrees and \[\dfrac{\pi }{2}\] radians in degree and radian units of angles respectively.