Convert The Following Angular Measures To Degrees, Minutes, And Seconds.a. $39.78^{\circ}$ B. $-32.98^{\circ}$ C. $ − 180 ∘ -180^{\circ} − 18 0 ∘ [/tex] D. $45.45^{\circ}$ E. $259.12^{\circ}$ F.
Introduction
Angular measures are an essential concept in mathematics, particularly in trigonometry and geometry. In this article, we will explore the process of converting angular measures from decimal degrees to degrees, minutes, and seconds. This conversion is crucial in various fields, including navigation, engineering, and architecture.
Understanding Angular Measures
Angular measures are typically expressed in decimal degrees, which represent the angle in terms of a fraction of a circle. For example, 39.78° is equivalent to 39 degrees and 46.68 minutes (since 0.78 × 60 = 46.8, which we round to 46.68 for simplicity). However, in many applications, it is more convenient to express angles in terms of degrees, minutes, and seconds.
Converting Decimal Degrees to Degrees, Minutes, and Seconds
To convert decimal degrees to degrees, minutes, and seconds, we can use the following steps:
- Multiply the decimal part of the angle by 60 to get the minutes.
- Multiply the decimal part of the minutes by 60 to get the seconds.
- Round the seconds to the nearest whole number.
Example 1: Converting 39.78° to Degrees, Minutes, and Seconds
Using the steps above, we can convert 39.78° to degrees, minutes, and seconds as follows:
- Multiply 0.78 by 60 to get 46.8 minutes.
- Multiply 0.8 by 60 to get 48 seconds.
- Round 48 seconds to the nearest whole number, which is 48 seconds.
Therefore, 39.78° is equivalent to 39° 46' 48".
Example 2: Converting -32.98° to Degrees, Minutes, and Seconds
Using the same steps, we can convert -32.98° to degrees, minutes, and seconds as follows:
- Multiply -0.98 by 60 to get -58.8 minutes.
- Multiply -0.8 by 60 to get -48 seconds.
- Since we are dealing with a negative angle, we need to add 60 minutes to the minutes to get a positive angle. Therefore, -58.8 minutes is equivalent to -58 minutes and 48 seconds.
- Therefore, -32.98° is equivalent to -32° 58' 48".
Example 3: Converting -180° to Degrees, Minutes, and Seconds
Since -180° is a whole number, we can express it directly in degrees, minutes, and seconds as follows:
- -180° is equivalent to -180° 0' 0".
Example 4: Converting 45.45° to Degrees, Minutes, and Seconds
Using the same steps, we can convert 45.45° to degrees, minutes, and seconds as follows:
- Multiply 0.45 by 60 to get 27 minutes.
- Multiply 0.5 by 60 to get 30 seconds.
- Therefore, 45.45° is equivalent to 45° 27' 30".
Example 5: Converting 259.12° to Degrees, Minutes, and Seconds
Using the same steps, we can convert 259.12° to degrees, minutes, and seconds as follows:
- Multiply 0.12 by 60 to get 7.2 minutes.
- Multiply 0.2 by 60 to get 12 seconds.
- Therefore, 259.12° is equivalent to 259° 7' 12".
Conclusion
Q: What is the purpose of converting angular measures from decimal degrees to degrees, minutes, and seconds?
A: Converting angular measures from decimal degrees to degrees, minutes, and seconds is essential in various fields, including navigation, engineering, and architecture. This conversion helps to provide more accurate and precise measurements, which is crucial in these fields.
Q: How do I convert a decimal degree to degrees, minutes, and seconds?
A: To convert a decimal degree to degrees, minutes, and seconds, follow these steps:
- Multiply the decimal part of the angle by 60 to get the minutes.
- Multiply the decimal part of the minutes by 60 to get the seconds.
- Round the seconds to the nearest whole number.
Q: What if the decimal part of the angle is negative?
A: If the decimal part of the angle is negative, you need to add 60 minutes to the minutes to get a positive angle. This is because a negative angle is equivalent to a positive angle that is 360° less.
Q: Can I convert a decimal degree to degrees, minutes, and seconds using a calculator?
A: Yes, you can convert a decimal degree to degrees, minutes, and seconds using a calculator. Most calculators have a built-in function to convert decimal degrees to degrees, minutes, and seconds.
Q: What is the difference between degrees, minutes, and seconds and decimal degrees?
A: Degrees, minutes, and seconds and decimal degrees are two different ways of expressing angular measures. Degrees, minutes, and seconds express the angle in terms of a fraction of a circle, while decimal degrees express the angle as a decimal value.
Q: Why is it important to round the seconds to the nearest whole number?
A: Rounding the seconds to the nearest whole number is important because it helps to provide a more accurate and precise measurement. This is especially important in fields where small errors can have significant consequences.
Q: Can I convert a decimal degree to degrees, minutes, and seconds using a spreadsheet?
A: Yes, you can convert a decimal degree to degrees, minutes, and seconds using a spreadsheet. Most spreadsheets have a built-in function to convert decimal degrees to degrees, minutes, and seconds.
Q: What if I have a decimal degree with a large number of decimal places?
A: If you have a decimal degree with a large number of decimal places, you can use a calculator or a spreadsheet to convert it to degrees, minutes, and seconds. Alternatively, you can use a conversion chart or a table to convert the decimal degree to degrees, minutes, and seconds.
Q: Can I convert a decimal degree to degrees, minutes, and seconds using a programming language?
A: Yes, you can convert a decimal degree to degrees, minutes, and seconds using a programming language. Most programming languages have a built-in function to convert decimal degrees to degrees, minutes, and seconds.
Conclusion
Converting angular measures from decimal degrees to degrees, minutes, and seconds is a crucial skill in various fields. By following the steps outlined in this article and using the FAQs provided, you can easily convert decimal degrees to degrees, minutes, and seconds. Remember to multiply the decimal part of the angle by 60 to get the minutes, and then multiply the decimal part of the minutes by 60 to get the seconds.