Rumata,
Conceptually you are not wrong to think that the order in which you do your sums should make any difference to the outcome.
If you used a calculator or computer program to do the sight reduction, which do not require a whole degree LHA, you could use your DR position unmodified in either method. Then either way you would have gotten the same LHA star and a calculated altitude and azimuth from your DR position.
conventional way:
GHA Aries 212° 05.7'
SHA star 146° 09.1'
GHA star 358° 14.8'
– DR lon 41° 50.0'
LHA star 316° 24.8'
unconventional way:
GHA Aries 212° 05.7'
– DR Long 41° 50.0'
LHA Aries 170° 15.7'
SHA star 146° 09.1'
LHA star 316° 24.8'
Here you can see that mathematically the sums are the same.
BUT
Your results differ because you have selected different longitude AP positions for each case. Neither of these are your actual position, but rather contrived to make the operation in which you apply them come out to a whole degree angle with no minutes and tenths.
You need the LHA of the star from your AP to be a whole degree to use the tables. That is because for the sake of compactness the tables only list solutions for navigation triangles with whole degree LHA's, not because the underlying math needs a whole degree angle LHA
In your second (unconventional) method you are you are trying to sum all the pertinent angles but you are not using the same angles as the first (conventional) method because you changed the AP. And while this unconventional AP results in a whole degree solution for your LHA Aries, it could never result in a whole degree solution for the LHA of the star unless the star happened to have a SHA with no minutes and tenths.
I hope this helps to clear things up for you.
Lastly, and this is just being nit-picky, doing the example the conventional way results in a difference of longitude from your DR to your AP greater than 30.0 arc minutes:
41° 50.0' – 41° 14.8' = 0° 35.2' LHA = 317°
You should update your AP longitude to 42° 14.8' W and then the LHA becomes 316°.
42° 14.8' – 41° 50.0' = 0° 24.8' LHA = 316°
Although either set of numbers will give you a legitimate LOP the second set of numbers will give you a shorter intercept and the answer that matches the exam, if you are taking one.
Conceptually you are not wrong to think that the order in which you do your sums should make any difference to the outcome.
If you used a calculator or computer program to do the sight reduction, which do not require a whole degree LHA, you could use your DR position unmodified in either method. Then either way you would have gotten the same LHA star and a calculated altitude and azimuth from your DR position.
conventional way:
GHA Aries 212° 05.7'
SHA star 146° 09.1'
GHA star 358° 14.8'
– DR lon 41° 50.0'
LHA star 316° 24.8'
unconventional way:
GHA Aries 212° 05.7'
– DR Long 41° 50.0'
LHA Aries 170° 15.7'
SHA star 146° 09.1'
LHA star 316° 24.8'
Here you can see that mathematically the sums are the same.
BUT
Your results differ because you have selected different longitude AP positions for each case. Neither of these are your actual position, but rather contrived to make the operation in which you apply them come out to a whole degree angle with no minutes and tenths.
You need the LHA of the star from your AP to be a whole degree to use the tables. That is because for the sake of compactness the tables only list solutions for navigation triangles with whole degree LHA's, not because the underlying math needs a whole degree angle LHA
In your second (unconventional) method you are you are trying to sum all the pertinent angles but you are not using the same angles as the first (conventional) method because you changed the AP. And while this unconventional AP results in a whole degree solution for your LHA Aries, it could never result in a whole degree solution for the LHA of the star unless the star happened to have a SHA with no minutes and tenths.
I hope this helps to clear things up for you.
Lastly, and this is just being nit-picky, doing the example the conventional way results in a difference of longitude from your DR to your AP greater than 30.0 arc minutes:
41° 50.0' – 41° 14.8' = 0° 35.2' LHA = 317°
You should update your AP longitude to 42° 14.8' W and then the LHA becomes 316°.
42° 14.8' – 41° 50.0' = 0° 24.8' LHA = 316°
Although either set of numbers will give you a legitimate LOP the second set of numbers will give you a shorter intercept and the answer that matches the exam, if you are taking one.