Exercise:

Alderney, in the Channel Islands, has longitude 2°W, latitude 50°N.
Winnipeg, in Canada, has longitude 97°W, latitude 50°N.
How far apart are they, in nautical miles, along a great circle arc?

Use the cosine rule:
cos AW = cos WP cos AP + sin WP sin AP cos P
= cos²40° + sin²40° cos 95°
= 0.5508
So AW = 56.58°
= 3395 nautical miles
(This is 7% shorter than the route along a parallel of latitude).

If you set off from Alderney on a great-circle route to Winnipeg,
in what direction (towards what azimuth) would you head?

Use the sine rule:
sin A / sin WP = sin P / sin WA
so sin x = sin 40° sin 95° / sin 56.58° = 0.77
so x = 50.1° or 129.9° .

Common sense says 50.1° (or check using cosine rule to get PW).

Azimuth is measured clockwise from north,
so azimuth is 360° - 50.1° = 309.9°

(Note that this is 40° north of the “obvious” due-west course.)

Back to "Spherical trigonometry".