arccos(.6) = 53.1301024 degrees
1 minute = 1/60 degrees
.1301024/(1/60) = 7.806144
or about 8 minutes if you round to the nearest minute
As a programmer, once I do the Cos-1(0.6) and get 53.13010235, I subtract the integer (53) and get 0.13010235. Then I multiply by 60 to get 7.806141234, which is 7 minutes. I then subtract the integer (7) to get 0.806141234. Then I multiply by 60 again to get the seconds, which is 48.36847404. This is how DMS conversion is done in your calculator. So the DMS answer is 53 deg 7' 48.36847404".
To get Decimal from the DMS (go backwards), I divide the 48.36847404" by 3600 which is 0.013435687. Then I take the minutes (7) and divide by 60 which is 0.116666667 and add both numbers to the 53 degrees:
0.013435687+0.116666667+53=53.13010235 which checks with the number we started with!