Question

Express in grades, minutes and seconds, the angles:
(i) ${{235}^{\circ }}12'36''$
(ii) ${{475}^{\circ }}13'48''$

Answer 1

Hint: First of all we will have to convert the given equation completely into degrees by using the conversion $1'={{\left( \dfrac{1}{60} \right)}^{\circ }}$ and $1''={{\left( \dfrac{1}{3600} \right)}^{\circ }}$.
After that we will use the following conversion to convert degrees into grades, minutes and seconds.
${{1}^{\circ }}=\dfrac{10}{9}grade$
${{1}^{\circ }}=60\operatorname{minutes}$
${{1}^{\circ }}=3600\operatorname{seconds}$

Complete step-by-step answer:
The given angles are expressed in grades, minutes and seconds as follows:
(i) ${{235}^{\circ }}12'36''$
First of all we will convert 12’ into degrees.
We know that $1'={{\dfrac{1}{60}}^{\circ }}$.
$\Rightarrow 12'={{\left( 12\times \dfrac{1}{60} \right)}^{\circ }}={{\left( \dfrac{12}{60} \right)}^{\circ }}={{\left( \dfrac{1}{5} \right)}^{\circ }}$
We also know that $1''={{\dfrac{1}{3600}}^{\circ }}$.

& \Rightarrow 36''={{\left( 36\times \dfrac{1}{3600} \right)}^{\circ }}={{\left( \dfrac{1}{100} \right)}^{\circ }} \\\ & \Rightarrow {{235}^{\circ }}12'36''={{235}^{\circ }}+12'+36'' \\\ \end{aligned}$$ Substituting the values of 12’ and 36’’ in terms of degree and then taking the LCM, we get as follows: $$\Rightarrow {{235}^{\circ }}12'36''=\dfrac{235\times 100+20+1}{100}={{\dfrac{23521}{100}}^{\circ }}$$ We know that $${{1}^{\circ }}=\dfrac{10}{9}grade$$ $$\Rightarrow {{\dfrac{23521}{100}}^{\circ }}=\left( \dfrac{23521}{100}\times \dfrac{10}{9} \right)=\dfrac{23521}{90}grade$$ Again, we know that $${{1}^{\circ }}=60\operatorname{minutes}$$. $$\Rightarrow \dfrac{{{23521}^{\circ }}}{100}=\left( \dfrac{23521}{100}\times 60 \right)\operatorname{minutes}=\dfrac{70563}{5}\operatorname{minutes}$$ We also know that $${{1}^{\circ }}=3600\operatorname{seconds}$$. $$\Rightarrow {{\left( \dfrac{23521}{100} \right)}^{\circ }}=\left( \dfrac{23521}{100}\times 3600 \right)\operatorname{seconds}=846756seconds$$ Hence $${{235}^{\circ }}12'36''$$ is equal to $$\dfrac{23521}{9}grade$$, $$\dfrac{70563}{5}\operatorname{minutes}$$, $$846756seconds$$. (ii) $${{475}^{\circ }}13'48''$$ First of all we will have to convert 13’ and 48’’ into degrees. Since we know that $$1'={{\dfrac{1}{60}}^{\circ }}$$. $$\Rightarrow 13'={{\left( 13\times \dfrac{1}{60} \right)}^{\circ }}={{\dfrac{13}{60}}^{\circ }}$$ We also know that $$1''={{\left( \dfrac{1}{3600} \right)}^{\circ }}$$. $$\begin{aligned} & \Rightarrow 48''={{\left( 48\times \dfrac{1}{3600} \right)}^{\circ }}={{\dfrac{1}{75}}^{\circ }} \\\ & \Rightarrow {{475}^{\circ }}13'48''={{475}^{\circ }}+13'+48'' \\\ \end{aligned}$$ On substituting the value of 13’ and 48’’ we get as follows: $$\Rightarrow {{475}^{\circ }}13'48''={{475}^{\circ }}+{{\dfrac{13}{60}}^{\circ }}+{{\dfrac{1}{75}}^{\circ }}={{\dfrac{475\times 300+13\times 5+4}{300}}^{\circ }}={{\dfrac{142500+65+4}{300}}^{\circ }}={{\dfrac{142569}{300}}^{\circ }}={{\dfrac{47523}{100}}^{\circ }}$$ Now we know that $${{1}^{\circ }}=\dfrac{10}{9}grade$$. $$\Rightarrow {{\left( \dfrac{47523}{100} \right)}^{\circ }}=\left( \dfrac{47523}{100}\times \dfrac{10}{9} \right)grade=\dfrac{15841}{30}grade$$ We already know that $${{1}^{\circ }}=60\operatorname{minutes}$$. $$\Rightarrow {{\left( \dfrac{47523}{100} \right)}^{\circ }}=\left( \dfrac{47523}{100}\times 60 \right)\operatorname{minutes}=\dfrac{14259}{5}\operatorname{minutes}$$ We already know that $${{1}^{\circ }}=3600\operatorname{seconds}$$. $$\Rightarrow {{\left( \dfrac{47523}{100} \right)}^{\circ }}=\dfrac{47523}{100}\times 3600\operatorname{seconds}=1710828seconds$$ Hence $${{475}^{\circ }}13'48''$$ is equal to $$\dfrac{15841}{30}grade$$, $$\dfrac{14259}{5}\operatorname{minutes}$$, $$1710828seconds$$. Therefore, the given angles are expressed in terms of grades, minutes and seconds as above. Note: Sometimes we use $${{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.