Problem 71 - Precalculus : Building Concepts and Connections

Question

Convert the angle measures given in DMS form to decimal degrees with three decimal places. $$16^{\circ} 24^{\prime} 45^{\prime \prime}$$

Step-by-Step Solution

Answer

The angle measure is \(16.413\) degrees, rounded to three decimal places.

Step 1: Identify the Degrees, Minutes, and Seconds

From the given angle measure in DMS form, which is \(16^{\circ} 24^{\prime} 45^{\prime \prime}\), the degrees are 16, the minutes are 24, and the seconds are 45.

Step 2: Convert Minutes and Seconds to Decimal Form

Convert the minutes to degrees by dividing by 60, and convert the seconds to degrees by dividing by 3600. This gives \(24/60 = 0.4\) degrees and \(45/3600 = 0.0125\) degrees.

Step 3: Add to Find Total Decimal Degrees

Now, add the degrees, the decimal equivalent of the minutes, and the decimal equivalent of the seconds together to get the total decimal degrees. This gives \(16 + 0.4 + 0.0125 = 16.4125\) degrees, rounded to three decimal places.

Key Concepts

DMS to decimal degrees

DMS stands for degrees, minutes, and seconds. It's a way to express angles similar to how time is expressed in hours, minutes, and seconds. To convert from DMS to decimal degrees, you essentially need to express the entire angle in terms of degrees as a single number. This involves remembering that:

• 1 minute (\(1'\)) is \(1/60^{\circ}\) of a degree.
• 1 second (\(1''\)) is \(1/3600^{\circ}\) of a degree.

The DMS representation, like \(16^{\circ} 24^{\prime} 45^{\prime \prime}\), is converted by adding all the parts: the degrees remain the same, while the minutes and seconds are converted into decimal degrees and added up. This is an essential skill in fields like geography, engineering, and aviation, where angles are commonly measured.

degree, minute, second conversion

The process of converting DMS to decimal degrees involves breaking down each component of the angle and expressing it in terms of degrees.Minutes to DegreesTo convert the minutes into degrees, divide the number of minutes by 60. In our exercise, we have 24 minutes, which translates to:\[24 \text{{ minutes}} = \frac{24}{60} = 0.4^{\circ}\]Seconds to DegreesSimilarly, convert seconds to degrees by dividing the number of seconds by 3600, since there are 3600 seconds in a full degree. For 45 seconds:\[45 \text{{ seconds}} = \frac{45}{3600} = 0.0125^{\circ}\]Finally, sum up all these values—degrees, converted minutes, and converted seconds—to get the total in decimal degrees.

decimal places in angle measurement

When converting angles to decimal degrees, precision is vital. The practice of rounding the answer to a specific number of decimal places—such as three—ensures that measurements are both accurate and easy to work with. Why Three Decimal Places? Three decimal places are often used in angle measurements to balance precision with practical usability. This level of detail is sufficient for many scientific and engineering calculations. For instance: An angle like 16.4125 degrees would be rounded to 16.413 degrees for clarity, while ensuring precision is maintained for calculations. Rounding the result in our exercise ensures our calculations remain close to the true value, avoiding unnecessary complexity that might arise from using too many decimal places. It's a simple but important step in converting DMS to decimal degrees, enabling consistent and reliable measurements.