Problem 69 - Precalculus : Building Concepts and Connections

Question

Convert the angle measures given in DMS form to decimal degrees with three decimal places. $$25^{\circ} 45^{\prime} 15^{\prime \prime}$$

Step-by-Step Solution

Answer

The converted value from DMS to decimal degrees is 25.754 degrees.

Step 1: Identifying the Parts of the DMS Angle

In the provided DMS angle, \(25^{\circ} 45^{\prime} 15^{\prime \prime}\), the degree part is 25, the minute part is 45, and the second part is 15.

Step 2: Converting the Minutes to Decimal Degrees

To convert the minutes to decimal degrees, divide the minutes (45) by 60. That means \(45 \div 60 = 0.75\).

Step 3: Converting the Seconds to Decimal Degrees

To convert the seconds to decimal degrees, divide the seconds (15) by 3600 (since there are 3600 seconds in one degree). So, \(15 \div 3600 = 0.00417\).

Step 4: Adding the Converted Parts Together

Now add up the decimal degree value of the original degrees, the converted minutes, and the converted seconds, to arrive at our final answer; so it is \(25 + 0.75 + 0.00417 = 25.75417\).

Key Concepts

Degree Measure

Degrees, minutes, and seconds are used to measure angles. The degree is the fundamental unit of this measurement, symbolized by \( ^{\circ} \). One full circle is divided into 360 degrees, making this unit very useful in geometry and navigation.
For instance, if you have an angle that measures \(25^{\circ}\), it means the angle is 25 degrees away from a reference line or point, like the start of a circle at zero degrees. Degrees are usually the largest measurement unit used in DMS notation, and all other units, minutes, and seconds, are fractions of a degree.

Minutes Conversion

Minutes serve as a subdivision of degrees in angle measurement. Just like how hours can be divided into smaller portions of minutes, degrees can be broken down into minutes for greater precision.
Each degree is divided into 60 minutes, noted as \(^{\prime}\). To convert minutes into a decimal degree, divide the number of minutes by 60. This is because there are 60 minutes in a single degree. For example, if you have 45 minutes, you calculate its decimal form as \( \frac{45}{60} = 0.75 \).
Thus, 45 minutes converted to degrees is 0.75 degrees.

Seconds Conversion

Seconds provide even finer granularity in measuring angles. One minute is further divided into 60 seconds. Therefore, seconds are the smallest unit in the DMS system, abbreviated with a double prime symbol \(^{\prime \prime}\).
To transform seconds into decimals of a degree, you divide the number of seconds by 3,600, since there are 3,600 seconds in a degree (60 seconds per minute times 60 minutes per degree).
Using 15 seconds as an example, you would calculate it as \( \frac{15}{3600} = 0.00417 \), providing a precise conversion for improved accuracy in measurements.

DMS to Decimal Form

Converting a DMS (degrees, minutes, seconds) angle into decimal form simplifies calculations and comparisons. To make this conversion, follow these steps:

• Keep the degree measure as is because it is already in decimal form.
• Convert minutes into a fraction of a degree by dividing by 60.
• Convert seconds into a fraction of a degree by dividing by 3,600.
• Add all these values together to get the final decimal degree.

To illustrate, take the angle \( 25^{\circ} 45^{\prime} 15^{ \prime \prime} \).
The degree stays 25, the minutes convert to 0.75, and the seconds convert to 0.00417. The complete decimal form becomes \( 25 + 0.75 + 0.00417 = 25.75417 \).
This precise form is often used in various fields, such as surveying and astronomy, where exact measurements are required.