Problem 34
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. $$330^{\circ} 25^{\prime \prime}$$
Step-by-Step Solution
Verified Answer
The decimal degree form of the given angle \(330^{\circ} 25^{\prime \prime}\) is approximately 330.007
1Step 1: Understand the Units
Before proceeding with any operation, it's important to understand what each unit indicates. A degree is divided into 60 arc minutes (represented by the prime symbol) and each arc minute is further divided into 60 arc seconds (represented by the double prime symbol). So, 1 degree equals 60 arc minutes or 3600 arc seconds.
2Step 2: Converting 25 Arc Seconds to Decimal Degree
Now, given 25 arc seconds need to be converted to decimal degree. To convert arcseconds into degree, divide the number of arcseconds by 3600 (since 1 degree equals to 3600 arcseconds). Doing so, we can calculate: 25 / 3600 = 0.006944444.
3Step 3: Adding Converted Value to The degree
After getting the converted value from arcseconds to degree, add this value to the given degree which is 330 in this case. Doing so, we get the final decimal degree as 330 + 0.006944444 = 330.006944444
Key Concepts
Decimal DegreesArc SecondsGraphing UtilityTrigonometric Conversion
Decimal Degrees
Decimal degrees are a way to express angles using decimals rather than traditional degrees, minutes, and seconds (DMS) format. This format is often used in various fields like geography, astronomy, and computer applications because it is simpler to use with calculators or computers.
In the DMS format, an angle might be expressed as something like \(330^{ ext{\circ}} 25''\). Here, 330 is the degree, and 25 is in arc seconds. To convert this to decimal degrees, you'll need to understand the conversions between units at different levels:
In the DMS format, an angle might be expressed as something like \(330^{ ext{\circ}} 25''\). Here, 330 is the degree, and 25 is in arc seconds. To convert this to decimal degrees, you'll need to understand the conversions between units at different levels:
- Degrees to decimal: Whole numbers remain the same.
- Minutes to degrees: Divide by 60.
- Seconds to degrees: Divide by 3600.
Arc Seconds
Arc seconds are smaller units of angular measurement used to provide precision. They are part of the degrees, minutes, and seconds system (DMS) and are used in fields requiring high precision like astronomy.
Here's how the breakdown works: There are 60 arc seconds in 1 arc minute and 3600 arc seconds in 1 degree. Therefore, arc seconds are very tiny, and converting them to a larger unit like degrees involves dividing by 3600. This is precisely what happens in the exercise where we have 25 arc seconds. You take this number, divide it by 3600, and you get approximately 0.006944444 degrees.
Here's how the breakdown works: There are 60 arc seconds in 1 arc minute and 3600 arc seconds in 1 degree. Therefore, arc seconds are very tiny, and converting them to a larger unit like degrees involves dividing by 3600. This is precisely what happens in the exercise where we have 25 arc seconds. You take this number, divide it by 3600, and you get approximately 0.006944444 degrees.
- Arc Seconds provide precision.
- Essential for accurate astronomical calculations.
Graphing Utility
A graphing utility is a digital tool designed to handle complex mathematical computations, including converting angle measures. Ideal for students and professionals alike, graphing utilities simplify tasks like converting between DMS format and decimal degrees.
When you input an angle, such as \(330^{\circ} 25''\), into a graphing calculator, it efficiently converts it using built-in functions. It follows the conversion rules between degrees, minutes, and seconds and provides an accurate decimal degree output.
When you input an angle, such as \(330^{\circ} 25''\), into a graphing calculator, it efficiently converts it using built-in functions. It follows the conversion rules between degrees, minutes, and seconds and provides an accurate decimal degree output.
- Speeds up computation.
- Reduces error in manual calculations.
- Eases the learning process for students.
Trigonometric Conversion
Trigonometric conversions are essential for various mathematical calculations involving angles. These conversions allow you to change angles between different units such as radians, degrees, and decimal degrees.
In trigonometry, angles are usually converted from degrees, minutes, and seconds to decimal degrees form because it simplifies the math involved. Once this is done, you can easily use trigonometric functions to solve different problems in fields like engineering, physics, and computer graphics.
In trigonometry, angles are usually converted from degrees, minutes, and seconds to decimal degrees form because it simplifies the math involved. Once this is done, you can easily use trigonometric functions to solve different problems in fields like engineering, physics, and computer graphics.
- Converts angles into usable forms.
- Facilitates the use of trigonometric functions.
- Reduces potential for calculation mistakes.
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