Chapter 4.1, Problem 82E

To determine

Convert $\text{0}\text{.7865 radian}$ into ${\text{D}}^{\text{0}}\text{M'}\text{S''}$ form.

Answer to Problem 82E

  ${45}^{0}3\text{'}32.4\text{'}\text{'}$

Explanation of Solution

Given: The angle $\text{0}\text{.7865 radian}$

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 radian and Degree $\text{1 radian = 57}\text{.29 deg}$

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

Relation between minutes and seconds $\text{1 minute = 60 seconds }$

Calculation:

  $\begin{array}{l}\text{1 radian = 57}\text{.29 deg}\\ \text{0}\text{.7865 radian = 0}\text{.7865 }·\text{ 57}\text{.29 deg = 45}{\text{.058585}}^{{}^0}\approx {45.059}^{{}^0}\end{array}$

  $\begin{array}{l}{45.059}^{{}^0}={45}^0+{0.059}^{{}^0}\\ 1^0=60\text{minutes}\\ \text{0}{\text{.059}}^0=0.059·60=3.54\text{minutes}\end{array}$  $\begin{array}{l}3.54\text{minutes}=3\min +0.54\min \\ 1\min =60\sec \\ 0.54\min =0.54·60=32.4\sec \end{array}$

  $\begin{array}{l}{45.059}^{{}^0}={45}^0+{0.059}^{{}^0}\\ {45.059}^{{}^0}={45}^0+3.54\text{minutes}\\ {45.059}^{{}^0}={45}^0+3\min +0.54\min \\ {45.059}^{{}^0}={45}^0+3\min +32.4\sec ={45}^{0}3\text{'}32.4\text{'}\text{'}\end{array}$

Thus, $\text{0}\text{.7865 radian}$ is equal to ${45}^{0}3\text{'}32.4\text{'}\text{'}$