To convert the angle $a = -81^\circ 31' 17''$ to decimal degrees, we'll need to convert both the minutes ($'$) and seconds ($''$) into a fraction of a degree. Here are the steps:

1. Convert seconds to minutes:

$17'' = \frac{17}{60} \, \text{minutes} \approx 0.28333'$

2. Add this to the minutes to get the total minutes as a decimal:

$31' + 0.28333' = 31.28333'$

3. Convert the total minutes to degrees:

$31.28333' = \frac{31.28333}{60} \, \text{degrees} \approx 0.5213883^\circ$

4. Add this to the whole degrees (keeping the negative sign):

$a = -81^\circ + 0.5213883^\circ = -81.5213883^\circ$

Conclusion:

The angle $a = -81^\circ 31' 17''$ in decimal degrees is approximately:

$a \approx -81.5214^\circ$

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Related Questions:

1. How do you convert decimal degrees back to degrees, minutes, and seconds?
2. What are some applications of using angles in decimal degrees?
3. How can you convert degrees to radians?
4. What is the significance of the negative sign in angle measurements?
5. How are angles measured differently in different coordinate systems?

Tip:

When converting minutes and seconds to decimal form, remember that there are 60 minutes in a degree and 60 seconds in a minute. This is key for accurate conversions!