Problem 77 - Precalculus : Building Concepts and Connections

Question

Convert the angle measures given in decimal degrees to DMS form. Round to the nearest second. $$40.25^{\circ}$$

Step-by-Step Solution

Answer

The angle \(40.25^{\circ}\) converted to DMS format is \(40^{\circ}15'0''\).

Step 1: Identify the degrees

The given angle is \(40.25^{\circ}\). The whole number part, 40, is the degree measure.

Step 2: Identify decimal part and convert it to minutes

The decimal part is .25 of one degree. We know that each degree contains 60 minutes, so \(.25\times60 = 15\) minutes.

Step 3: Calculate the remaining decimal part in seconds

The decimal part after multiplying by 60 is zero, thus no further calculations are required.

Step 4: Write the angle in DMS format

The angle is now fully converted and can be written in DMS format as 40 degrees, 15 minutes, and 0 seconds, denoted as \(40^{\circ}15'0''\).

Key Concepts

Degrees Minutes Seconds (DMS)

Degrees, Minutes, Seconds (DMS) is a way to express angles using degrees, minutes, and seconds. This form is particularly useful in fields such as navigation and astronomy.

• Degrees are the main unit of angle measurement. One full circle is 360 degrees.
• Each degree is divided into 60 minutes (denoted by an apostrophe, for example, 15').
• Each minute is further divided into 60 seconds (denoted by double apostrophes, such as 30'').

To convert an angle from one unit to another within the DMS system, you multiply or divide by 60. For example, if you have 0.5 of a degree, you multiply by 60 to find the minutes, which would be 30 minutes. Understanding DMS format is essential for precise angle measurement, especially in professions that require detailed geographic or celestial positioning.

Decimal Degrees

Decimal Degrees is a simpler way to express angles where the degree values include a decimal fraction. This format is most commonly used in geographic information systems (GIS) and digital mapping.

• It combines degrees and fractions of a degree into a single number.
• Decimal degrees make calculations easier because they don’t require conversions between different units like minutes and seconds.
• For example, 40.25° directly tells you the angle is more than 40° and a quarter past 40° (or 15 minutes in DMS).

Converting decimal degrees to DMS (and vice versa) is a common task for students and professionals alike. A key advantage of decimal degrees is the straightforwardness in addition and subtraction, as it avoids the complexity of working in base 60.

Angle Measurement

Angle Measurement is an essential concept in geometry, trigonometry, and many applied sciences. Angles can be measured in different units, each useful depending on the context.

• Degrees are the most familiar unit and often used in everyday contexts such as carpentry and engineering.
• Radians are used extensively in higher-level mathematics and physics; one full circle equals 2π radians.
• DMS and decimal degrees are both variations of degree measurement, differing mainly in their presentation.

Knowing how to convert between these units allows for greater flexibility in problem-solving. Regardless of the unit, the fundamental concept of an angle remains the same: it is a way to measure the rotation necessary to bring one line, ray, or segment into coincidence with another.