Problem 20 - Precalculus with Calculus Previews

Question

In Problems \(17-20,\) express the given angle in decimal notation. $$ 10^{\circ} 25^{\prime} $$

Step-by-Step Solution

Answer

The angle is approximately 10.4167° in decimal notation.

Step 1: Understand the Problem

We are given an angle in degrees and minutes notation, specifically \(10^{\circ} 25^{\prime}\). The goal is to convert this angle into decimal notation, which involves expressing the angle in terms of degrees only, including fractions of a degree.

Step 2: Convert Minutes to Degrees

Since 1 degree equals 60 minutes, each minute is \(\frac{1}{60}\) of a degree. Therefore, to convert 25 minutes into degrees, we compute \(\frac{25}{60}\). This calculation gives us the decimal equivalent of the minutes in degrees.

Step 3: Perform the Calculation

Calculate \(\frac{25}{60}\) to find out how many degrees 25 minutes is:\[\frac{25}{60} = 0.4167\]So, 25 minutes is equal to approximately 0.4167 degrees.

Step 4: Add to Degrees

Now add the converted minutes (in decimal form) to the initial degree measure:\[10 + 0.4167 = 10.4167\]This result represents the angle in decimal notation.

Key Concepts

Degrees and Minutes

Angles can be represented in different ways, and one common method is using degrees and minutes. A degree is a unit of angle measurement, and there are 360 degrees in a complete circle. Each degree is divided into 60 minutes. It's similar to how hours and minutes work on a clock.
For instance, an angle might be given as \(10^{\circ} 25^{\prime}\). Here, \(10^{\circ}\) represents 10 whole degrees and \(25^{\prime}\) stands for 25 minutes. This format provides a precise way to describe angles, especially helpful in fields like navigation and astronomy. The goal in many exercises is to convert this format into a more easily computable decimal notation, which involves a simple conversion process.

Decimal Notation

Decimal notation for angles is a streamlined way to express angle measurements using decimal numbers. Unlike degrees and minutes, decimal notation expresses the smaller divisions of a degree directly in decimal form. For instance, when converting \(10^{\circ} 25^{\prime}\) into decimal notation, we end up with a single number: \(10.4167\).
Using decimal notation simplifies calculations, as it allows for angles to be easily added, subtracted, multiplied, or divided without needing to consider the conversions between minutes and degrees.
To achieve this conversion, the minutes must be translated into a fraction of a degree. This involves dividing the number of minutes by 60 since there are 60 minutes in a degree. The result gives us the decimal part that we add to the whole degrees.

Fractional Degrees

Fractional degrees are crucial when converting from degrees and minutes to decimal notation. Every minute constitutes a fraction of a degree. Specifically, each minute represents \(\frac{1}{60}\) of a degree.
To convert 25 minutes into degrees, you calculate \(\frac{25}{60}\), resulting in approximately 0.4167 degrees. This fraction must then be added to the total number of whole degrees to obtain the final angle in decimal form.
This process helps in expressing angles more accurately and is particularly useful for calculations that require precision. It's important to note that, while working with fractional degrees, you often round the final result to four decimal places for practical purposes, ensuring accuracy and consistency in computations.