Question

How do you convert the angle $85{}^\circ 18'30''$ in decimal degree form?

Answer 1

Hint: In this question, we have to convert the angle into the degree. So, we will apply the degree-angle formula to get the solution for the same. First, we will convert the 18 minute into degree by using the formula ${1}'={{\left( \dfrac{1}{60} \right)}^{\circ }}$ , and then convert 30 seconds into minute using the formula ${1}''=\text{ }{{\left( \dfrac{1}{60} \right)}^{\prime }}$ . After finding the minute value, we will again convert it into the degree using the same formula ${1}'={{\left( \dfrac{1}{60} \right)}^{\circ }}$. After that, we will add the degree and new degree to get the required solution to the problem.

Complete step-by-step answer:
According to the question, we have to find the value of the degree by converting the angle into degree form.
So, we will use the angle-degree formula.
The angle-minute-seconds given to us is $85{}^\circ 18'30''$ --------- (1)
As we know, 1 degree is equal to 60 minute, that is
\[{{1}^{\circ }}=60\min =6{0}'\]
$\Rightarrow {1}'={{\left( \dfrac{1}{60} \right)}^{\circ }}$ ---------- (2)
So, we will change the 18 minute into degree, that is we will multiply 18 on both sides in the equation (2), we get
$\Rightarrow {1}'.(18)=18.\left( {{\dfrac{1}{60}}^{\circ }} \right)$
$\Rightarrow 1{8}'={{\dfrac{18}{60}}^{\circ }}$
$\Rightarrow 1{8}'={{\dfrac{18}{60}}^{\circ }}$
Therefore, we get
$\Rightarrow 1{8}'={{\dfrac{3}{10}}^{\circ }}={{0.3}^{\circ }}$ -------------- (3)
Also, we know that 1 minute is equal to 60 seconds, that is
${1}'=6{0}''$
$\Rightarrow {1}''=\text{ }{{\left( \dfrac{1}{60} \right)}^{\prime }}$ ----------- (4)
So, now we will change 30 seconds into minute, that is multiply 30 on both sides in the equation (4), we get
$\begin{align}
  & \Rightarrow {1}''.(30)={{\left( \dfrac{1}{60} \right)}^{\prime }}.(30) \\
 & \Rightarrow 3{0}''={{\dfrac{30}{60}}^{\prime }} \\
\end{align}$
Therefore, we get
$\Rightarrow 3{0}''={{\dfrac{1}{2}}^{\prime }}$ ----------- (5)
Also, we know that 1 degree is equal to 60 minute, that is
\[1{}^\circ =60\min =6{0}'\]
$\Rightarrow {1}'=\text{ }{{\left( \dfrac{1}{60} \right)}^{{}^\circ \text{ }}}$ ---------- (6)
So, now we will change $\dfrac{1}{2}$ minute into degree, that is multiply $\dfrac{1}{2}$ on both sides in the equation (6), we get
\[\begin{align}
  & \Rightarrow {1}'.\left( \dfrac{1}{2} \right)={{\left( \dfrac{1}{60} \right)}^{{}^\circ }}.\left( \dfrac{1}{2} \right) \\
 & \Rightarrow {{\left( \dfrac{1}{2} \right)}^{\prime }}={{\left( \dfrac{1}{2.(60)} \right)}^{{}^\circ }} \\
\end{align}\]
\[\Rightarrow {{\left( \dfrac{1}{2} \right)}^{\prime }}={{\left( \dfrac{1}{120} \right)}^{{}^\circ }}={{0.008333333}^{\circ }}\] ----------- (7)
Therefore, we now add the values of equation (3) and (7) we get
$\Rightarrow 1{8}'3{0}''={{0.3}^{\circ }}+{{0.008333333}^{\circ }}$
Therefore, we get
$\Rightarrow 1{8}'3{0}''={{0.308333333}^{\circ }}$ ------------ (8)
Now, we will add 30 degree and equation (8), we get
$\Rightarrow {{30}^{\circ }}1{8}'6{0}''={{30}^{\circ }}+{{0.308333333}^{\circ }}$
Thus, we get
$\Rightarrow {{30}^{\circ }}1{8}'6{0}''={{30.308333333}^{\circ }}$
Therefore, for the angle $85{}^\circ 18'30''$ , the decimal degree is ${{30.308333333}^{\circ }}$ .

Note: While finding the degree form from the angle-minute-seconds form, keep in mind that to convert minute into the degree, 1 minute is equal to $\dfrac{1}{60}$ degree and when we convert second to degree, 1 second is equal to $\dfrac{1}{3600}$ degree. Do mention the formula in the steps to avoid the error.