TR-04Z: Degrees Minutes and Seconds (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial will guide you through the concept of measuring angles using Degrees, Minutes, and Seconds (DMS) notation, a vital skill in trigonometry. Understanding DMS is essential for various applications in mathematics, engineering, and navigation. By the end of this guide, you’ll be able to convert between DMS and decimal degrees confidently.

Step 1: Understanding DMS Notation

Degrees, Minutes, and Seconds are units used to represent angles. Here’s how they break down:

• Degrees: The largest unit, where 360 degrees make a full circle.
• Minutes: Each degree is divided into 60 minutes (denoted as ').
• Seconds: Each minute is further divided into 60 seconds (denoted as '').

Practical Tip

Always remember that:

• 1 degree = 60 minutes
• 1 minute = 60 seconds

Step 2: Converting Decimal Degrees to DMS

To convert a decimal degree to DMS, follow these steps:

1. Extract the whole number: This is the degree part.
2. Multiply the decimal by 60: This gives you the minutes.
   • Take the decimal part from step 1, multiply it by 60.
3. Extract the whole number: This is the minutes part.
4. Multiply the new decimal by 60: This gives you the seconds.
5. Round the seconds: If necessary, round to the desired precision.

Example

Convert 34.567 degrees to DMS:

1. Degrees: 34
2. Minutes: 0.567 * 60 = 34.02 → 34 minutes
3. Seconds: 0.02 * 60 = 1.2 → 1.2 seconds (round to 1 second)

Final result: 34° 34' 1"

Step 3: Converting DMS to Decimal Degrees

To convert from DMS to decimal degrees:

1. Take the degrees: This remains as is.
2. Divide the minutes by 60: Convert minutes to degrees.
3. Divide the seconds by 3600: Convert seconds to degrees (since 3600 seconds = 1 degree).
4. Add them together: Sum all parts for the final decimal degree.

Example

Convert 45° 30' 15" to decimal degrees:

1. Degrees: 45
2. Minutes: 30 / 60 = 0.5
3. Seconds: 15 / 3600 = 0.00416667
4. Total: 45 + 0.5 + 0.00416667 = 45.50416667

Final result: 45.5042 degrees

Step 4: Practice with Examples

To reinforce your understanding, practice converting between DMS and decimal degrees using the following examples:

1. Convert 12° 15' 30" to decimal degrees.
2. Convert 75.25 degrees to DMS.
3. Convert 90° 0' 0" to decimal degrees.
4. Convert 180.5 degrees to DMS.

Conclusion

Understanding DMS notation is crucial for accurately measuring and converting angles in various fields. Remember to practice the conversions to gain confidence. For your next learning step, consider exploring the introduction to radians in the upcoming video. Happy learning!