Problem 73 - Precalculus : Building Concepts and Connections

Question

Convert the angle measures given in DMS form to decimal degrees with three decimal places. $$120^{\circ} 50^{\prime} 15^{\prime \prime}$$

Step-by-Step Solution

Answer

The angle \(120^{\circ} 50^{\prime} 15^{\prime \prime}\) in decimal degrees, rounded to three decimal places, is \[120.837^{\circ}\].

Step 1: Identify Degrees, Minutes, and Seconds

In the problem, the angle measure is given as \(120^{\circ} 50^{\prime} 15^{\prime \prime}\). Here, the DMS factors can be identified as follows: Degrees = 120, Minutes = 50, Seconds = 15.

Step 2: Convert Minutes and Seconds to Decimal Degrees

Once the degrees, minutes, and seconds have been identified, they can be converted into decimal degrees using the formula: Decimal Degrees = Degrees + (Minutes/60) + (Seconds/3600). So, apply this formula with the given values.

Step 3: Perform the calculation

Plug the identified values into the formula. Decimal Degrees = 120 + (50 / 60) + (15 / 3600). Always remember to divide the minutes by 60 and the seconds by 3600 to convert them to degrees. Do the calculations.

Step 4: Round the Decimal Degrees

The exercise instructs to give the final result in decimal degrees with three decimal places. After performing the calculation in step 3, round the result to three decimal places.

Key Concepts

Angle Measures

Angle measures describe the size of an angle, which is the space between two intersecting lines or surfaces at, or close to, the point where they meet. Angles are commonly measured in degrees, which are full circle divided into 360 parts known as degrees.
There are different ways to express angle measures, including DMS (Degrees, Minutes, Seconds) and decimal degrees. These two formats allow us to understand angles in various contexts, like navigation, geometry, and astronomy.
Understanding how to manipulate and convert between these formats is crucial for many applications that require precise angle measurement and adjustments.

Degrees Minutes Seconds

When describing angles, the Degrees Minutes Seconds (DMS) format is a common choice. In this format, the measure is broken down into three components:

• Degrees (\(^\circ\)) represent the whole number of degrees.
• Minutes (\(\prime\)) are a subdivision of degrees, with 1 degree equaling 60 minutes.
• Seconds (\(\prime\prime\)) are a further subdivision, where 1 minute equals 60 seconds.

Writing DMS is akin to using hours, minutes, and seconds for telling time, providing a precise way of denoting angles. For example, the angle measure seen as \(120^{\circ} 50^{\prime} 15^{\prime\prime}\) is interpreted as 120 degrees, 50 minutes, and 15 seconds, offering a high degree of precision beyond just degrees.

Decimal Degrees Conversion

To understand how to convert DMS to decimal degrees, we need to a simplify the components into a single number. The formula revolves around dividing the minutes by 60 and the seconds by 3600:\[ \text{Decimal Degrees} = \text{Degrees} + \left(\frac{\text{Minutes}}{60}\right) + \left(\frac{\text{Seconds}}{3600}\right) \]This methodology allows us to incorporate the smaller subdivisions of minutes and seconds as fractions of a degree. Converting \(120^{\circ} 50^{\prime} 15^{\prime\prime}\) results in:

• The degrees stay as they are: 120
• The 50 minutes are converted by dividing by 60, resulting in 0.8333
• The 15 seconds are divided by 3600, resulting in approximately 0.0042

Add these together to get the final decimal degrees:\[ 120 + 0.8333 + 0.0042 = 120.8375 \]When rounding this to three decimal places, we obtain 120.838. This conversion simplifies angle measurements and is often used in fields requiring precise calculations, like computer graphics and geolocation applications.