Destination Longitude from Course and Distance

This gives the longitude reached after running a distance $d$ along the great circle that leaves a start at latitude $\varphi_1$ and longitude $\lambda_1$ on an initial true bearing $\theta$. It is the second half of a destination point, the first half being the latitude $\varphi_2$ from the library's destination-latitude page; here $\varphi_2$ is found inside the page from the same start, bearing and distance. The distance enters as the angle $\delta = d/R$ at the centre of the Earth. The change in longitude is the angle of a point whose two coordinates are $\sin\theta\,\sin\delta\,\cos\varphi_1$ and $\cos\delta - \sin\varphi_1\sin\varphi_2$; $\mathrm{atan2}$ uses the signs of both to place that angle in the right quadrant, so the result has the right sign as well as the right size.

Watch the sign convention. Ed Williams' Aviation Formulary counts longitude positive to the west, so it subtracts the change from the starting longitude. This page counts longitude positive to the east, as the rest of the library does, so it adds the change instead: $\lambda_2 = \lambda_1 + \mathrm{dlon}$, where dlon is the $\mathrm{atan2}$ term, wrapped into the range from -180 to 180 degrees. Latitude is north positive and the bearing is clockwise from true north.

The bearing is the initial one: along a great circle the course changes as you go, except along the equator and the meridians. Those two exceptions make good checks. Due north along a meridian does not change the longitude, which the formula shows because $\sin\theta$ is zero for a bearing of 0; the page's example, 1 000 km due north from 40 degrees north on 20 degrees east, returns 20 degrees. Due east along the equator from the Greenwich meridian for a quarter of the Earth's circumference, $\pi R / 2$ or 10 007 557.22 m, returns 90 degrees.

The Formulary's own worked example is a waypoint 100 nautical miles out of Los Angeles (33 degrees 57 minutes north, 118 degrees 24 minutes west) on the 66 degree radial (1.150035 radians). It lands at 34 degrees 37 minutes north and 116 degrees 33 minutes west. The page gives -116.5514 degrees, and run backwards from that waypoint it returns the start longitude of -118.4000 degrees. It uses the mean Earth radius of 6 371 008.8 m.