Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Convert each angle measure to degrees, minutes, and seconds. Use a calculator as necessary. Round to the nearest second. $$31.4296^{\circ}$$

Step-by-Step Solution

Answer

31.4296° is equivalent to 31° 25' 47''.

Step 1: Separate Whole Degrees from Decimal

The angle given is \(31.4296^{\circ}\). First, separate the whole number part, which is \(31^{\circ}\), from the decimal part, which is \(0.4296\).

Step 2: Convert Decimal to Minutes

Take the decimal part \(0.4296\) and convert it to minutes. Multiply by 60:\[0.4296 \times 60 = 25.776\] This means there are 25 minutes and 0.776 of a minute.

Step 3: Convert Remaining Decimal to Seconds

Take the remaining decimal part of minutes \(0.776\) and convert it to seconds. Multiply by 60 again:\[0.776 \times 60 = 46.56\] This means there are approximately 47 seconds after rounding to the nearest second.

Step 4: Combine Degrees, Minutes, and Seconds

Combine the whole degrees, the converted minutes, and the converted seconds to express the angle in degrees, minutes, and seconds form:\[31^{\circ} 25' 47''\]

Key Concepts

Degrees, Minutes, and Seconds

When dealing with angles, expressing them in degrees, minutes, and seconds is a common practice. This format is similar to how we express time, with 1 degree being analogous to an hour. Here:

• 1 degree (°) is divided into 60 minutes (')
• 1 minute (') is divided into 60 seconds ('')

Using this system, angles can be expressed more precisely. For example, an angle that is less than a full degree can still be accurately described down to a fraction of a second. This method allows us to perform calculations and convey measurements with great precision, which is crucial in fields such as astronomy and surveying.

Decimal to Minutes Conversion

Converting decimal degrees to minutes is an important step when working with angle measurements. This process involves:

• First identifying the whole number of degrees, which remains unchanged.
• The remaining decimal is then converted to minutes by multiplying it by 60.

For instance, if we have an angle of 31.4296°, we separate the 31° from the decimal 0.4296. To find how many minutes this decimal represents, multiply by 60: 0.4296 × 60 = 25.776 minutes. This tells us that there are 25 full minutes and some fraction of a minute remaining. Converting decimals to minutes helps us move closer to a full degree, enabling more precise notation.

Calculator Usage for Angles

Using a calculator can greatly simplify the process of converting angles from decimal degrees to degrees, minutes, and seconds. Here’s how to effectively use a calculator for these tasks:

• Ensure your calculator is in degree mode, as some calculations may require switching between radians and degrees.
• Use multiplication operations to convert the decimal part of degrees to minutes by multiplying by 60.
• Repeat the multiplication to convert any remaining decimal from minutes to seconds.

Calculators are handy as they quickly and accurately handle the conversions and rounding, saving time and reducing errors in manual calculations. Understanding how to operate the calculator in this context is a valuable skill when dealing with angle conversions.

Rounding to Nearest Second

Rounding is a crucial step in expressing angles in degrees, minutes, and seconds, as it allows for a clean representation of the data. After converting decimals, rounding helps align the measurement with its closest second for practical use. Here's how you do it:

• Convert the decimal from minutes to seconds by multiplying by 60, resulting in a decimal number of seconds.
• Look at the fraction of seconds. If it's 0.5 or greater, round up to the next whole second.
• If it's less than 0.5, round down, keeping the integral part as is.

In the example with 31.4296°, once the conversion to seconds gives us 46.56, we round it to 47 seconds. Accurate rounding is key for precision and clarity in conveying angular measurements.