Problem 80 - Precalculus : Building Concepts and Connections

Question

Convert the angle measures given in decimal degrees to DMS form. Round to the nearest second. $$120.68^{\circ}$$

Step-by-Step Solution

Answer

The angle measure in DMS form is \(120^{\circ} 40' 48''\).

Step 1: Find Whole Degrees

The whole number part of the decimal degree gives the degree portion of the DMS. So, the given angle \(120.68^{\circ}\) has 120 degrees.

Step 2: Compute Minutes

To calculate the minutes, first subtract the whole degree from the total degree. Then, multiply the decimal part by 60. Doing this calculation: \(60*(0.68)=40.8\) The whole number part gives 40 minutes.

Step 3: Compute Seconds

After finding the minutes, subtract the whole number of minutes from the decimal minute obtained in previous step. Then, multiply the decimal part by 60 to find the number of seconds. Thus: \(60*(0.8)=48\) which means there are 48 seconds. As per the problem's instruction, we are rounding to the nearest second.

Key Concepts

Angle Measurement Conversion

Understanding how to convert between different systems of angle measurement is a fundamental skill in fields such as astronomy, geography, and various branches of engineering. Just as we can move between metric and imperial units for length, we can convert angle measurements from decimal degrees to the degrees, minutes, and seconds (DMS) format.

When starting with a decimal degree, like in the example of 120.68°, the whole number represents the degrees. Converting the decimal part involves a straightforward multiplication process. By multiplying the decimal by 60, we convert it into minutes, since there are 60 minutes in a degree. We then repeat this step with the remaining decimal to convert it into seconds, because there are also 60 seconds in a minute. This step-by-step approach allows for a seamless transition between the formats, ensuring precision and ease of understanding.

Degrees Minutes Seconds (DMS) Format

The DMS format is an alternative to decimal degrees for representing angles, relying on a sexagesimal (base 60) system which dates back to ancient Sumerians and Babylonians. This historical notation divides each degree into 60 minutes, and each minute into 60 seconds. While it might seem more complex than modern decimal degrees, DMS is particularly useful in fields where precision is paramount, or when geographical coordinates are being displayed.