Question

Convert each angle to \(D^{\circ} M^{\prime} S^{\prime \prime}\) form. Round your answer to the nearest second. \(40.32^{\circ}\)

Short answer

40° 19' 12''

Step-by-step solution

Separate the degrees

Start by identifying the whole number part of the given angle. In this case, it's 40. This is the degrees part, so we have 40 degrees.

Calculate the minutes

Take the decimal part of the angle, which is 0.32 in this case. Multiply this by 60 to convert it into minutes: \[0.32 \times 60 = 19.2\]So, we have 19 minutes.

Calculate the seconds

Take the remaining decimal part from the previous step (0.2 minutes) and multiply by 60 to convert it into seconds: \[0.2 \times 60 = 12\]So, rounding to the nearest second, we have 12 seconds.

Combine parts

Combine all the parts: 40 degrees, 19 minutes, and 12 seconds. So, the angle in \(D^{\circ} M^{\text{\prime}} S^{\text{\prime\prime}}\) form is: \[40^{\circ} 19^{\prime} 12^{\prime\prime}\].

Key concepts

degrees to minutes conversion

To convert degrees to minutes, start by isolating the decimal part of your degree measurement. For example, if you have 40.32 degrees, separate it into 40 degrees and 0.32 degrees. This 0.32 degrees is what you will convert into minutes.
Since 1 degree equals 60 minutes, you can use the following calculation:
0.32 degrees × 60 = 19.2 minutes.
So, 0.32 degrees equals 19.2 minutes.

Remember to always multiply the decimal part of the degree value by 60.
This conversion helps in breaking down the angle into more manageable parts for further calculation.

minutes to seconds conversion

Converting minutes to seconds is a crucial step when working with angles. After you have converted your degrees to minutes, you may still have a decimal part. For example, from 19.2 minutes, you can isolate 0.2 minutes for further conversion.

To convert this decimal part (0.2 minutes) to seconds, multiply it by 60 because 1 minute equals 60 seconds:
0.2 minutes × 60 = 12 seconds.
So, 0.2 minutes equals 12 seconds.

This step ensures that your angle measurement is as precise as possible, breaking it down to the nearest second.

rounding to nearest second

When you're converting angles, precision is key. Sometimes, after converting minutes to seconds, you might get a value that isn't a whole number. In such cases, rounding to the nearest second becomes important.

For instance, if your calculation yields 12.4 seconds, you would round it to 12 seconds because 12.4 is closer to 12 than to 13. However, if you get 12.6 seconds, you'd round it to 13 seconds because 12.6 is closer to 13.

This way, you ensure that your final answers are not only precise but also easier to understand.
Always consider rounding when the second part of your conversion does not come out as a whole number.