Problem 31 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the angle-conversion capabilities of a graphing utility to convert the angle measure to decimal degree form. Round your answer to three decimal places, if necessary. $$85^{\circ} 18^{\prime} 30^{\prime \prime}$$

Step-by-Step Solution

Answer

The angle \( 85^{\circ} 18^{\prime} 30^{\prime \prime}\) in decimal form is \( 85.308 \) degrees.

Step 1: Convert min to Degree

First, convert the minutes to degrees. Given that 1 degree is equivalent to 60 minutes, we can convert 18 minutes into degrees by dividing 18 by 60. That is \( \frac{18}{60} = 0.3 \) degrees.

Step 2: Convert sec to Degree

Next, convert the seconds to degrees. Given that 1 degree is equivalent to 3600 seconds, convert 30 seconds into degrees by dividing 30 by 3600. That is \( \frac{30}{3600} = 0.0083 \) degrees.

Step 3: Sum up to get final Value

Now, sum up the decimal values obtained with the given degrees to obtain the angle measure in decimal form. That is: \( 85 + 0.3 + 0.0083 = 85.3083 \) degrees.

Key Concepts

Decimal Degree Conversion

Understanding decimal degree conversion is essential for various fields including navigation, surveying, and in particular, mathematics. Converting an angle given in degrees, minutes, and seconds to a decimal format allows us to express the angle in a more compact and sometimes more usable form.

For instance, consider the angle given as \(85^{\text{o}} 18' 30''\). To convert this to decimal degrees, we need to convert each part - degrees, minutes, and seconds - separately into decimal format and then combine them.

• Since there are 60 minutes in one degree, we divide the number of minutes by 60 to get the degree equivalent.
• Similarly, there are 3600 seconds in one degree, so dividing the number of seconds by 3600 gives us their decimal degree equivalent.

Combining these values gives us the angle in decimal degrees, which is a more fluid representation widely used in digital applications and trigonometry.

Degrees to Minutes Conversion

Degrees and minutes are units used to measure angles, and they come from a historical background where divisions were made in a base-60, or sexagesimal, system. This is why we have 60 minutes in one degree and 3600 seconds in one degree.

To convert angles from degrees to minutes, you need to multiply the degree part by 60. If the angle already has minutes, just add them to the product. This conversion plays a role when you're dealing with precision tasks that require a more granular measurement than whole degrees can provide. For example, the decimal degree of 0.3, when converted back to minutes, equals \(0.3 \times 60 = 18\) minutes. This process is the reverse of the decimal degree conversion and is equally crucial for many practical applications, including trigonometry and astronomy.

Trigonometry Graphing Utilities

In the age of technology, trigonometry graphing utilities offer a robust platform for visualizing and understanding complex trigonometric concepts. These utilities serve as both pedagogic tools for students and computational aids for professionals.

Graphing utilities are particularly useful for converting angles between different units of measure, plotting trigonometric functions, and exploring their properties. They allow users to input an angle in degrees, minutes, and seconds and obtain a decimal degree value, or vice versa. This is done through internal algorithms - similar to the steps shown in the exercise - which automate the conversion process and eliminate manual calculation errors.

• Graphing utilities handle large datasets to create accurate graphs.
• They include features for zooming and adjusting scales to observe minute details of trigonometric graphs.
• One can also overlay multiple functions for comparative study, which is particularly useful in educational settings.

These features enhance the learning and application of trigonometry, making graphing utilities indispensable tools in this digital era.