Problem 74 - Precalculus : Building Concepts and Connections

Question

Convert the angle measures given in DMS form to decimal degrees with three decimal places. $$150^{\circ} 40^{\prime} 20^{\prime \prime}$$

Step-by-Step Solution

Answer

The converted angle measure is approximately 150.672 degrees.

Step 1: Understanding Degrees Minutes and Seconds

Degrees, minutes and seconds is a system of measurement for angles. It is represented as \(a^{\circ} b^{\prime} c^{\prime \prime}\), where \(a\) is the degrees, \(b\) is the minutes and \(c\) is the seconds. 1 degree includes 60 minutes and 1 minute has 60 seconds.

Step 2: Setup the conversion

For converting the angle to decimal degrees, the given degrees \(a\) is left as it is. The minutes \(b\) must be divided by 60 to convert it to degrees. Similarly, the seconds \(c\) must be divided by 3600 for conversion. These values are then added together to get the final decimal degrees value.

Step 3: Perform Conversion

Start with the given degrees, minutes, and seconds: \(150^{\circ}, 40^{\prime}\), and \(20^{\prime \prime}\). Carry out the division for the conversion and add the values: \( 150 + \frac{40}{60} + \frac{20}{3600}\). It gives approximately 150.672 decimal degrees.

Key Concepts

Degrees, Minutes, and Seconds

The system of measuring angles using degrees, minutes, and seconds ( DMS ) has been around for a long time. This method expresses an angle in terms of degrees ( ^{ circle} ), minutes ( ^{ circle} ), and seconds ( '' ), where there are

• 60 minutes in a degree
• 60 seconds in a minute

To better understand, consider the angle 150° 40' 20'' which means: 150 degrees, 40 minutes, and 20 seconds. Each unit serves a distinct purpose: the degree is the largest unit indicating the overall size of the angle, while minutes and seconds offer a means to measure smaller subdivisions within a degree.
Knowing how to read and interpret DMS is crucial as it is frequently used in fields that require precise angle measurements, such as astronomy and navigation.

Decimal Degrees

Decimal degrees offer a simplified alternative to the Degrees, Minutes, and Seconds system of angle measurement. Instead of breaking an angle into parts, decimal degrees (DD) express these measurements as a single real number. This streamlines calculations and comparisons.
To convert from DMS to DD, each component (degrees, minutes, and seconds) of an angle must be transformed and combined:

• The degree value remains unchanged.
• The minute component is divided by 60 as there are 60 minutes in a degree.
• The second component is divided by 3600 as there are 3600 seconds in a degree.

For example, converting 150° 40' 20'' involves using the formula:\[150 + \frac{40}{60} + \frac{20}{3600} = 150.672\]This method enables precise and straightforward representations of angles which benefits computational operations and aids in software implementations reliant on angle measurements.

Measurement Systems

Throughout history, various measurement systems for angles have been developed to suit different needs and contexts. The two primary systems that persist today are the Degrees, Minutes, and Seconds (DMS), and Decimal Degrees (DD). Each has its applications and advantages:

• **DMS** : The traditional system useful for detailed, fractional measurements. It is vital in fields requiring high precision, such as navigation and certain engineering applications.
• **DD** : Modern and convenient for a wide range of scientific and technical purposes. It simplifies calculations, especially those involving geographic information systems (GIS) and computer graphics.

Choosing between these systems often depends on the specific requirements of the task at hand—whether a detailed fraction of a degree is more beneficial than a simple decimal notation. Both enable the measurement of angles accurately but provide flexibility in how the information is formatted and processed.