Question

Express in grades, minutes and seconds, the angles:
(i) \[{{30}^{\circ }}\]
(ii) \[{{81}^{\circ }}\]

Answer 1

Hint: For the above question, we will have to know about the conversion of angles into grades, minutes and seconds which is given as follows:
In grades \[{{1}^{\circ }}=\dfrac{10}{9}grade\]
In minutes \[{{1}^{\circ }}=60\operatorname{minutes}\]
In seconds \[{{1}^{\circ }}=3600\operatorname{seconds}\]

Complete step-by-step answer:
We will express the given angle in grades, minutes and seconds as follows:
(i) \[{{30}^{\circ }}\]
We know that \[{{1}^{\circ }}=\dfrac{10}{9}grade\]
\[\Rightarrow {{30}^{\circ }}=30\times \dfrac{10}{9}grade=\dfrac{300}{9}grade=32.2grade\] (approx.)
We know that \[{{1}^{\circ }}={{60}^{\circ }}\operatorname{minutes}\].
\[\Rightarrow {{30}^{\circ }}=30\times 60\operatorname{minutes}=1800minutes\]
Also, we know that \[{{1}^{\circ }}=3600\operatorname{seconds}\].
\[\Rightarrow {{30}^{\circ }}=30\times 3600\operatorname{seconds}=108000seconds\]
Hence, \[{{30}^{\circ }}=32.2grade=1800\operatorname{minutes}=108000seconds\].
(ii) \[{{81}^{\circ }}\]
We know that \[{{1}^{\circ }}=\dfrac{10}{9}grade\]
\[\Rightarrow {{81}^{\circ }}=\left( 81\times \dfrac{10}{9} \right)grade=90grade\]
Once again, we know that \[{{1}^{\circ }}=60\operatorname{minutes}\].
\[\Rightarrow {{81}^{\circ }}=81\times 60\operatorname{minutes}=4860minutes\]
Also, we know that \[{{1}^{\circ }}=3600\operatorname{seconds}\].
\[\Rightarrow {{81}^{\circ }}=\left( 81\times 3600 \right)\operatorname{seconds}=291600\]
Therefore the given angles are expressed in grades, minutes and seconds as above.

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.