Problem 5 - Precalculus

Question

Convert the angles into decimal degrees. Round each of your answers to three decimal places. $$ 125^{\circ} 50^{\prime} $$

Step-by-Step Solution

Answer

125° 50′ converts to 125.833 decimal degrees rounded to three decimal places.

Step 1: Understand the Notation

The angle given is written as \(125^{\circ} 50^{\prime}\), which means 125 degrees and 50 minutes. Our task is to convert this into decimal degrees.

Step 2: Conversion of Minutes to Degrees

We know that 1 degree is equal to 60 minutes. Thus, to convert minutes to decimal degrees, we use the formula:\[ \text{Decimal Degrees} = \frac{\text{Minutes}}{60} \]For 50 minutes, the conversion is:\[ \frac{50}{60} = 0.8333 \] (rounded to four decimal places for accurate intermediate calculation)

Step 3: Add Decimal Degrees to Whole Degrees

Now add the decimal degrees you calculated from the minutes to the whole degrees:\[ 125 + 0.8333 = 125.8333 \]

Step 4: Round the Result

Round the result to three decimal places as required by the exercise:\[ 125.8333 \approx 125.833 \]

Key Concepts

Decimal Degrees

Decimal degrees are a way of expressing angles using only degrees and decimal fractions of a degree, rather than using the traditional degrees, minutes, and seconds (DMS) format. In decimal degrees, the minutes and seconds are converted into a decimal format. This is particularly useful in mathematics and science, where calculations are simplified without the need for additional conversions.
To convert an angle from degrees and minutes to decimal degrees, you use the relationship that 1 degree = 60 minutes. For example, if you have 50 minutes, you convert it by dividing 50 by 60, which gives the decimal equivalent of 0.8333 degrees (rounded to four decimal places for precision during calculations). This decimal part can then be added to the whole degrees to get the angle in decimal form. For example, an angle of 125 degrees 50 minutes becomes 125.8333 decimal degrees before final rounding to three places.

Rounding Numbers

Rounding numbers is a numerical method used to make calculations simpler while maintaining a level of precision that is sufficient for the task at hand. It involves either increasing or decreasing a number to a specified degree of accuracy.
Here we use rounding to ensure that the final angle measurement is in a simpler form, such as rounding to three decimal places. In the example conversion of 125 degrees 50 minutes, after converting to decimal degrees, we obtained an initial result of 125.8333. To round this to three decimal places, you look at the fourth decimal digit. Since it is 3, which is less than 5, you keep the third decimal digit at 3.

• If the next digit is 5 or more, you increase the last considered digit by one.
• If the next digit is less than 5, you leave the last considered digit.

This way, your result becomes 125.833.

Degree and Minute Notation

Degree and minute notation is a traditional way of measuring angles. It allows for high precision by using smaller units known as minutes.
In this notation:

• One degree is subdivided into 60 minutes (0 lines and ′ symbols are used).
• Traditionally, each minute can be further divided into 60 seconds, though often, calculations stop at minutes to keep things simple.

For example, an angle could be expressed as 125° 50′. This displays the angle with 125 full degrees and an additional 50 minutes. This is particularly useful in fields like cartography and astronomy, where precise angle measurements are crucial.
To convert this into a simpler decimal degree format, each component (like the 50 minutes) is converted into its decimal degree equivalent using the relationship 1 degree = 60 minutes, resulting in a unified angle representation.