Chapter 4.1, Problem 69E

To determine

Convert ${85}^{0}18\text{'}30\text{'}\text{'}$ into decimal degree form.

Answer to Problem 69E

  ${85.308}^0$

Explanation of Solution

Given: The angle ${85}^{0}18\text{'}30\text{'}\text{'}$

Concept Used:

Degree is a measure of an angle. One degree is one 360th part of a full circle.

Each degree is split up into 60 parts, each part being 1/60 of a degree. These parts are called minutes.

Each minute is split up into 60 parts, each part being 1/60 of a minute. These parts are called seconds.

One degree is 60 minutes. One minute is 60 seconds. 360 degrees is the same as a complete circle around north-south axis or from the equator.

Relation between degree and minute $60\text{minutes}=1^0$

Relation between Degrees and seconds $3600\sec =1^0$

Calculation:

  $\begin{array}{l}60\text{minutes}=1^0\\ 1\text{minute}\text{=}\frac{1}{60}\mathrm{deg}\\ 18\min \text{ute}s=18·\frac{1}{60}\mathrm{deg}=\frac{18}{60}\mathrm{deg}=\frac{3}{10}\mathrm{deg}=0.3\mathrm{deg}\\ 3600\sec =1^0\\ 1\sec =\frac{1}{3600}\mathrm{deg}\\ 30\sec =30·\frac{1}{3600}\mathrm{deg}=\frac{30}{3600}\mathrm{deg}=\frac{1}{120}\mathrm{deg}=0.0083\mathrm{deg}\\ \\ {85}^{0}18\text{'}30\text{'}\text{'}={85}^0+18\text{'}+30\text{'}\text{'}={85}^0+{0.3}^0+{0.0083}^0={85.30833}^0\approx {85.308}^0\end{array}$

Thus, ${85}^{0}18\text{'}30\text{'}\text{'}$ is equal to ${85.308}^0$