Every spot on Earth has a latitude (how far north or south of the equator) and a longitude (how far east or west of Greenwich). The shortest path between two places follows a great circle, and the distance is just degrees of separation times 111.2 km.
On a meridian, the angular distance is the difference in latitude. On a parallel, it's the difference in longitude, and θ is that latitude.
Two buoys on the 150°E meridian, at 20°S and 50°S. How far apart?
• cos θ only for the same parallel (same latitude). On a meridian there's no cos θ.
• Same hemisphere = subtract, opposite hemispheres = add.
• On a parallel, θ is the latitude of that parallel.
• Distance is in kilometres, the angular distance is in degrees.
Parallels (latitude) run east to west; meridians (longitude) run north to south. A and B share a meridian, so the distance between them uses 111.2 × the latitude difference.
Circles of latitude get smaller as you move away from the equator. At 60° the circle is half the equator's size, and cos 60° = 0.5, so the same gap in degrees covers half the distance. That's exactly what the cos θ does.
Don't read yet, just have a go in your head:
20°S to 50°S: both south, subtract → 30°. D = 111.2 × 30 = 3336 km.
12°N to 48°N on a meridian: angular = 48 − 12 = ?° → D = 111.2 × ? = ?
Two stations on the 30°N parallel, at 20°E and 50°E. Find the distance. (Use cos θ, θ = 30°.) Check below.
Two places on the same meridian are 2780 km apart. What is the angular distance between them?
In one sentence, out loud: when do you include cos θ, and what is θ?