Problem 21 - Precalculus: Functions and Graphs

Question

Exer. \(21-24:\) Express the angle as a decimal, to the nearest ten-thousandth of a degree. $$37^{\circ} 41^{\prime}$$

Step-by-Step Solution

Answer

The angle is approximately \(37.6833^{\circ}\).

Step 1: Understand the Components

Angles are often expressed in degrees, minutes, and seconds. In this exercise, you are given the angle as \(37^{\circ} 41^{\prime}\), where \(37^{\circ}\) represents degrees and \(41^{\prime}\) represents minutes.

Step 2: Convert Minutes to Decimal Degrees

1 minute is equal to \( \frac{1}{60} \) of a degree. Thus, the minutes can be converted to degrees by multiplying the minutes by \( \frac{1}{60} \). So, convert \(41^{\prime}\) to degrees: \[ 41^{\prime} = 41 \times \frac{1}{60} = 0.6833^{\circ} \]

Step 3: Sum Degrees and Minutes

Add the degrees part and the decimal degrees converted from minutes to get the total angle in decimal form:\[ 37^{\circ} + 0.6833^{\circ} = 37.6833^{\circ} \]

Step 4: Round to the Nearest Ten-Thousandth

The computed angle was already rounded to the nearest ten-thousandth in the previous step, which is \(37.6833^{\circ}\).

Key Concepts

Degrees, Minutes, and Seconds

When measuring angles, the most common unit is degrees, denoted by the degree symbol °. However, angles can also be measured using smaller units called minutes and seconds.

• A degree is divided into 60 minutes, similar to how an hour is divided into minutes in time.
• Each minute is further divided into 60 seconds.

This system is particularly useful for precision and is similar to the way we handle time. An angle might be expressed, for instance, as "37 degrees, 41 minutes," just like we express time as "37 hours and 41 minutes." This system allows for a fine-tuned measure of angles which is crucial in fields like navigation, astronomy, and even some engineering tasks. Understanding this will help in converting these units to other forms like decimal degrees.

Decimal Degrees

Decimal degrees are a simpler way to represent angles without using the minutes and seconds. In this format, angles are expressed as a single decimal number.

• This is convenient for calculations, as it eliminates the need to work with multiple units (degrees, minutes, and seconds).
• Decimal degrees are particularly useful in digital technology applications like GPS, where calculations and data storage benefit from the simplified format.

The conversion from degrees, minutes, and seconds to decimal degrees involves transforming each minute and second into a fraction of a degree. This format allows for easy use in various computational settings and helps in ensuring precision.

Rounding to Ten-Thousandth

When dealing with decimal degrees, rounding becomes important to ensure that numbers do not become unnecessarily long, which might complicate interpretations.

• The ten-thousandth place is the fourth digit to the right of the decimal point.
• Rounding to the nearest ten-thousandth involves a specific method: you look at the fifth digit. If it's 5 or greater, the fourth digit increases by one; if it's less than 5, the fourth digit stays the same.

This level of precision is usually more than adequate for most scientific and engineering uses. It balances the need for accuracy with the practicality of using and communicating numbers efficiently.

Mathematical Conversion

Converting from degrees, minutes, and seconds to decimal degrees involves some straightforward calculations. Here's a simple technique for understanding this conversion:

• First, keep the degrees as they are. This is your principal component.
• Next, convert the minutes by dividing by 60, since there are 60 minutes in a degree.

For example, if you have 41 minutes, you convert it to decimal degrees by calculating 41 divided by 60, which equals approximately 0.6833. This calculation can be repeated similarly for any quantity of seconds by dividing by 3600 (60 minutes and 60 seconds make up one degree). These conversions are vital in fields requiring precise angle measurements, enhancing understanding and accuracy in applications like geodesy and astronomy.