Okay, so here's how the calculation should have gone:
acos((sin(21°34') - sin(8°27.4') · sin(54°41')) / (cos(8°27.4') · cos(54°41')))
acos((0.367583568 - 0.147061367 · 0.815969464) / (0.98912737 · 0.578095005))
acos((0.367583568 - 0.119997585) / 0.571809592)
acos(0.247585984 / 0.571809592)
acos(0.432986762) = 64.34274255 or 64°20.6'
If we subtract the GHA of the Sun from the LHA, we get a longitude of 4°22.6' E. This is also clearly wrong. I did check your data for the Sun on the date you gave and it is correct. I'm not sure how you initially got a value of 0.82, though.
Anyway, you wrote that you "measured that sun angle in a slightly rough sea so if the result is a bit different than expected that's the reason why". That's perfectly normal under the conditions. However, if we calculate what the altitude of the Sun should have been at that time and location, we get:
asin(sin(8°27.4) · sin(54°41') + cos(8°27.4) · cos(79°02') · cos(54°41')) = 13°13.5'
That's a difference of 8°20.5' from your measured altitude. This is far too much error for an accurate determination of longitude (or for any navigational purpose). An error of a few tenths of an arcminute would be acceptable. Even a measurement error of one whole arcminute at that latitude would result in an error in longitude of a little more than 1 nautical mile.
(Note also: the farther off your estimated latitude is from your actual latitude, the greater the error in longitude will be. However the closer the Sun is to due East or West when the sight is taken, the smaller this error will be. But I don't think that is the problem here. The azimuth of the Sun at the time of your sight should have been 266° - almost due West.)
In summary: the only problem I can find is in the measured altitude of the Sun. What the cause of this error is, I cannot say.
|