To convert angles from the DMS (degrees, minutes, and seconds) format to the centesimal measure, we follow a detailed step-by-step process:
1. Convert the DMS to decimal degrees:
[tex]\[
\text{Decimal Degrees} = \text{Degrees} + \left(\frac{\text{Minutes}}{60}\right) + \left(\frac{\text{Seconds}}{3600}\right)
\][/tex]
2. Convert the decimal degrees to centesimal measure:
[tex]\[
\text{Centesimal Measure} = \text{Decimal Degrees} \times \left(\frac{10}{9}\right)
\][/tex]
Let's apply this process to each given angle:
### i) 37° 55' 22"
Step 1: Convert to decimal degrees
[tex]\[
\text{Decimal Degrees} = 37 + \left(\frac{55}{60}\right) + \left(\frac{22}{3600}\right)
\][/tex]
[tex]\[
\text{Decimal Degrees} = 37 + 0.9167 + 0.0061 = 37.9228 \, (\text{approximately})
\][/tex]
Step 2: Convert to centesimal measure
[tex]\[
\text{Centesimal Measure} = 37.9228 \times \left(\frac{10}{9}\right) = 42.1364 \, (\text{approximately})
\][/tex]
So, 37° 55' 22" in centesimal measure is 42.1364.
### ii) 24° 25' 37"
Step 1: Convert to decimal degrees
[tex]\[
\text{Decimal Degrees} = 24 + \left(\frac{25}{60}\right) + \left(\frac{37}{3600}\right)
\][/tex]
[tex]\[
\text{Decimal Degrees} = 24 + 0.4167 + 0.0103 = 24.4270 \, (\text{approximately})
\][/tex]
Step 2: Convert to centesimal measure
[tex]\[
\text{Centesimal Measure} = 24.4270 \times \left(\frac{10}{9}\right) = 27.1410 \, (\text{approximately})
\][/tex]
So, 24° 25' 37" in centesimal measure is 27.1410.
### iii) 36° 30' 45"
Step 1: Convert to decimal degrees
[tex]\[
\text{Decimal Degrees} = 36 + \left(\frac{30}{60}\right) + \left(\frac{45}{3600}\right)
\][/tex]
[tex]\[
\text{Decimal Degrees} = 36 + 0.5000 + 0.0125 = 36.5125 \, (\text{approximately})
\][/tex]
Step 2: Convert to centesimal measure
[tex]\[
\text{Centesimal Measure} = 36.5125 \times \left(\frac{10}{9}\right) = 40.5694 \, (\text{approximately})
\][/tex]
So, 36° 30' 45" in centesimal measure is 40.5694.
Thus, the conversions of the given angles to centesimal measure are:
i) 37° 55' 22" = 42.1364
ii) 24° 25' 37" = 27.1410
iii) 36° 30' 45" = 40.5694
