Problem 23 - Precalculus: Functions and Graphs

Question

Exer. \(21-24\) : Express the angle as a decimal, to the nearest ten-thousandth of a degree. $$115^{\circ} 26^{\prime} 27^{\prime \prime}$$

Step-by-Step Solution

Answer

The angle is approximately \(115.4408^{\circ}\) to the nearest ten-thousandth.

Step 1: Understand the Angle Measurement

Angles can be expressed in degrees, minutes, and seconds. This is similar to how time is expressed as hours, minutes, and seconds. In this exercise, we have an angle of \(115^{\circ} 26' 27''\) that needs to be converted to a decimal form, which is purely in degrees.

Step 2: Convert Minutes to Decimal Degrees

One degree is equal to 60 minutes. To convert 26 minutes to degrees, we use the conversion formula: \[ 26' = \frac{26}{60}^{\circ} \approx 0.4333^{\circ} \].

Step 3: Convert Seconds to Decimal Degrees

Similarly, one minute is equal to 60 seconds. To convert 27 seconds to degrees, we use the conversion formula: \[ 27'' = \frac{27}{3600}^{\circ} \approx 0.0075^{\circ} \].

Step 4: Add All Contributions

Add the degrees, the converted minutes, and the converted seconds to obtain the total angle in decimal degrees: \[ 115^{\circ} + 0.4333^{\circ} + 0.0075^{\circ} = 115.4408^{\circ} \].

Step 5: Round to the Nearest Ten-Thousandth

The calculated decimal degrees is already to the nearest ten-thousandth. Therefore, the final angle expressed as a decimal to the nearest ten-thousandth is \(115.4408^{\circ}\).

Key Concepts

Decimal Degrees

Decimal degrees are a way to express angles using decimal points to represent the parts of a degree, rather than minutes and seconds. This method can simplify mathematical calculations. Instead of dealing with degrees, minutes, and seconds, the angle is expressed as a single decimal number. For example, an angle of \(115^{\circ} 26' 27''\) in degrees, minutes, and seconds can be converted into a decimal degree format for ease of use, like \(115.4408^{\circ}\).

This format is particularly useful because:

• It facilitates easy computation and addition/subtraction of angles.


• It provides a consistent format for inputting angles into digital systems and software programs that may not accept minutes or seconds.


• It allows for simple representation and interpretation in scientific and engineering tasks.

Understanding how to convert between these formats is vital, especially in fields like geography, astronomy, and navigation.

Degrees Minutes Seconds

The Degrees, Minutes, and Seconds (DMS) is a traditional way of expressing angles, much like how time is divided into hours, minutes, and seconds.

Within this system:

• Degrees (\(^{\circ}\)) are the largest unit.


• One degree is comprised of 60 minutes ('), symbolized by a prime.


• Each minute is made up of 60 seconds (''), indicated by a double prime.

When you encounter an angle such as \(115^{\circ} 26' 27''\), it means 115 whole degrees, with an additional 26 minutes and 27 seconds.

This method of expressing angles is common in navigation and astronomy. But, to use it in mathematical calculations, we often convert it into decimal degrees, as performing arithmetic operations directly with DMS can be cumbersome.

Angle Measurement

Angle measurement is a fundamental concept in mathematics that describes the rotation or opening between two lines converging at a point, called the vertex. It is usually measured in degrees (\(^{\circ}\)) or radians. Degrees are more common in everyday applications, like navigation and geometrics, making it essential to be familiar with different angle formats.

The first step in understanding angle measurement is recognizing how angles are expressed:

• Back in the day, angles were mainly articulated using the Degrees, Minutes, and Seconds (DMS) system.


• Nowadays, both DMS and decimal degrees are widely used, with the latter being preferred in computational contexts.

By converting between these two, you can ensure accurate angle calculations across various applications, from setting up satellite dishes to performing in complex trigonometric computations.

Precalculus

Precalculus is a mathematical course that bridges the gap between algebra and calculus, covering various foundational concepts, including angle conversions. Within precalculus, understanding how angles can be expressed and converted is crucial.

This subject serves multiple roles:

• It prepares students for the rigorous problem-solving processes found in calculus.


• It lays down the principles of trigonometry, including how to convert between Degrees, Minutes, Seconds (DMS) and decimal degrees.


• It introduces students to analytic geometry, aiding in the understanding of how angles function in graphs and shapes.

By grasping angle conversion, students can navigate through more complex topics in precalculus that will later support their understanding of calculus and other higher-level math topics.