Problem 31 - Basic Technical Mathematics with Calculus

Question

Change the given angles to equal angles expressed to the nearest minute. $$-5.62^{\circ}$$

Step-by-Step Solution

Answer

The angle -5.62° is -5° 37'.

Step 1: Understand the Problem

We need to convert the decimal degree (-5.62°) into degrees and minutes, rounded to the nearest minute. One degree is composed of 60 minutes.

Step 2: Separate the Decimal Part

Identify the integer part of the angle, which is -5°, and separate the decimal part, which is 0.62°.

Step 3: Convert the Decimal to Minutes

Multiply the decimal part by 60 to convert it to minutes. Calculate: 0.62° × 60 = 37.2 minutes.

Step 4: Round Minutes to the Nearest Whole Number

Round the calculated minutes (37.2) to the nearest whole number, which gives us 37 minutes.

Step 5: Compile the Final Answer

Combine the integer degrees and the rounded minutes to express the angle. The converted angle is -5° 37'.

Key Concepts

Understanding Decimal Degrees

Decimal degrees are a way of expressing angles using decimal fractions rather than traditional degrees, minutes, and seconds (DMS). This format is especially handy in many scientific and mapping applications because it allows for easier computation.
- A whole number represents the degrees. - The decimal part represents a fraction of a degree, which can be further converted into minutes and seconds. For instance, in the angle \(-5.62^{\circ}\), the \-5^{\circ}\ represents the whole degree, and \(.62^{\circ}\) is the fractional part that can be converted into minutes and seconds. The key to converting decimal degrees properly is to treat the decimal as a part of the degree. Multiply it by 60 to convert it to minutes, as there are 60 minutes in each degree. This method keeps things simple and consistent across different calculations.

Converting to Degrees and Minutes

Converting decimal degrees into degrees and minutes is a straightforward task once you know the basics. Simply separate the integer part from the decimal in the decimal degree notation.
Let’s go through the conversion process of \-5.62^{\circ}\:

• First, identify the integer part, \-5^{\circ}\, which remains the same.
• Next, take the decimal \(.62^{\circ}\), multiply it by 60 to convert into minutes: \(.62 \times 60 = 37.2 \, \text{minutes}\).

After calculating the minutes, combine them back with the integer degree value. Converting decimal degrees to degrees and minutes is all about recognizing that each decimal fraction corresponds to a part of 60 minutes. Always remember, the deeper you go into precision, the more accurate your representation of the angle will be in terms of smaller subdivisions.

Rounding Angles to Nearest Minute

Rounding angles to the nearest minute is crucial for many practical applications. After converting from decimal degrees, rounding helps ensure you have a clean, usable value.
When you obtain a result like 37.2 minutes, you need to decide the nearest whole number, which represents the rounded minutes. Here’s a simple way to determine:

• If the decimal part is 0.5 or higher, round up.
• If the decimal part is less than 0.5, round down.

So, in the case of \(-5^{\circ} 37.2'\), since 0.2 is less than 0.5, you would round to \(-5^{\circ} 37'\). Rounding ensures your angle is easily interpreted and stays within standardized measurement conventions, avoiding precision loss that might happen with more specific figures like seconds.