Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Express \(\theta\) in terms of degrees, minutes, and seconds, to the nearest second. $$\theta=1.5$$

Step-by-Step Solution

Answer

\(\theta = 1^\circ 30' 0''\)

Step 1: Understand the Conversion Scale

We are given an angle \(\theta\) in decimal degrees and need to convert it to degrees, minutes, and seconds. There are 60 minutes in a degree and 60 seconds in a minute.

Step 2: Convert Decimal to Degrees

The whole part of the decimal, which is 1, represents full degrees. Therefore \(1.5\) implies 1 degree and a decimal part of 0.5 degrees remaining.

Step 3: Convert Remaining Decimal to Minutes

To convert the decimal part (0.5) to minutes, multiply by 60 (since there are 60 minutes in a degree): \(0.5 \times 60 = 30\). This means there are 30 minutes.

Step 4: Convert Decimal Part of Minutes to Seconds

Since 0.5 has been fully converted to 30 minutes with no decimal remaining, the number of seconds is 0. Thus, there isn't a need for further conversion of a fractional part to seconds.

Step 5: Compile the Final Answer

We have 1 degree, 30 minutes, and 0 seconds. Hence, \(\theta = 1^\circ 30' 0''\) in degrees, minutes, and seconds.

Key Concepts

Decimal Degrees

Decimal degrees are a way to express angles using decimal numbers rather than fractions. This method is often used in geographic coordinate systems, where precision and simplicity are key. In decimal degrees, the whole number represents the degrees, while the decimal portion expresses a fraction of a degree.

Understanding decimal degrees is crucial for converting angles into more traditional formats, such as degrees, minutes, and seconds, which are used in various applications, like navigation and astronomy. For example, if you have a decimal degree value of 1.5, it means you have one full degree plus half of another degree. Converting this to degrees, minutes, and seconds allows for a more intuitive understanding of how much of the circle or sphere is covered by the angle.

Degrees Minutes Seconds

Degrees, minutes, and seconds (DMS) is a traditional format used to represent angles. Each degree is divided into 60 minutes, and each minute is further divided into 60 seconds. This system is akin to how we measure time, making it familiar and useful in many practical contexts.

In converting angles from decimal degrees to DMS format, you start by identifying the whole degree value, then use the decimal portion to determine the corresponding minutes and seconds. For example, 1.5 in decimal degrees translates to 1 degree and 0.5 of a degree. By converting the 0.5 degree into minutes, you multiply by 60 (since there are 60 minutes in a degree), resulting in 30 minutes. If any decimal portion of the minutes remains, it would be further multiplied by 60 to find the seconds. This system is essential in fields requiring precise angle measurements.

Conversion Scale

A conversion scale provides a straightforward method to convert angles from decimal degrees to the degrees, minutes, and seconds format. Understanding this scale is imperative because it serves as the roadmap to transition between these numerical systems.

The conversion steps follow a sequence:

• Identify the whole number for complete degrees.
• Multiply the remaining decimal by 60 to convert it into minutes.
• If there is any remaining decimal after converting to minutes, multiply it again by 60 to convert it to seconds.

For instance, converting 1.5 decimal degrees, we take the 1 as the whole number of degrees, and convert the 0.5 to 30 minutes. As there is no needs for additional conversion to seconds in this case, we get the final DMS result. This conversion scale is practical and can be used for any decimal degree value for efficient transformation.

Educational Mathematics

Educational mathematics often involves converting numbers between different forms to deepen understanding and application of mathematical concepts. This exercise of angle conversion from decimal degrees to degrees, minutes, and seconds highlights the importance of understanding both the decimal system and traditional measurement systems.

For students, mastering these conversions provides valuable skills that are applicable in various fields, such as geography, astronomy, and engineering. It also enhances numerical literacy, encouraging flexibility in thinking about numbers and their uses. Educational resources focused on these concepts aim to simplify the conversion process with clear steps and real-world examples, ensuring learners not only know how to perform conversions but also understand why they are useful, thereby elevating their mathematical competence and confidence.