Following a great circle
This is the "direct" problem of geodesy: given a start, an initial bearing θ and a distance d, find the end point. On a sphere of radius R, with δ = d / R as the angle travelled:
φ₂ = asin(sin φ₁ · cos δ + cos φ₁ · sin δ · cos θ); λ₂ = λ₁ + atan2(sin θ · sin δ · cos φ₁, cos δ − sin φ₁ · sin φ₂)
The main answer comes from Karney's solution of the direct problem on the WGS84 ellipsoid; the sphere's answer is shown underneath with the gap between the two. Over 10 km they differ by a few tens of metres; over thousands of kilometres, by kilometres.
The traveller holds the starting bearing only for a moment, then follows the great circle, so the final bearing is usually different. Start on the equator heading 090° and you stay on it; head north and you cross the pole and come back down the other side, as the "12,000 km north" example shows. Uses include dead reckoning, laying out a search area or checking a waypoint someone described as "12 km on 210° from the summit".