Initial Great Circle Bearing

This is the course to steer at the instant of departure, and only at that instant. On any great circle other than a meridian or the equator, the track crosses each successive meridian at a different angle, so the bearing changes continuously the whole way along. That is not a defect of the calculation. It is precisely what makes the great circle shorter than the constant-course alternative, and the two facts cannot be separated.

Treating an initial bearing as a course to hold is the classic error of the subject and it is worth being specific about the damage. Departing New York for London on 051° and holding 051° does not put you in London; it puts you well to the south of it, because the great circle bends north of the constant-bearing track and then comes back down. On a high-latitude crossing the divergence runs to tens of miles within a few hours. The working practice, from Bowditch onward, is to break the great circle into legs — commonly every 5° of longitude, or every few hours of running — and recompute this bearing at each waypoint. What you are actually sailing is a chain of short rhumb lines that approximates the great circle, which is exactly what a chart plotter's route function does for you now.

The two-argument arctangent matters more than it looks. Using a plain $\arctan(y/x)$ collapses the four quadrants into two and returns a bearing 180° wrong for half of all possible destinations — the answer looks perfectly reasonable, which is what makes it dangerous. The function used here keeps the signs of both arguments and then the result is wrapped into 0° to 360°, because that is the range a compass card, a chart and a controller all speak in. A bearing of −130° is a programmer's answer, not a navigator's.

Finally, this is a TRUE bearing: measured from true north, which is where the meridians on the chart point. It is not a compass course and it is not a magnetic course. Before anyone steers it, variation for the place and deviation for the vessel have to be applied, and in the air a wind correction angle goes on before either of those. Each of those has its own page here, and the order they are applied in is not arbitrary.