The problem by the St. Hilaire method resolves itself into the solution of the spherical triangle shown in Fig. 10, where two sides and an included angle are given:
Polar distance = 90° - declination (Nautical Almanac).
Co-latitude = 90° - latitude (by dead reckoning).
Hour angle of star. See figure and solution below.
The hour angle of the sun is more readily found than that of a star. It is accomplished by applying the longitude (in time) of the assumed position to the Greenwich time shown by the chronometer at the time of sight. This hour angle of the mean sun must be corrected by the equation of time to obtain the hour angle of the actual sun.
Fig. 9.
Fig. 10.
The cosine-haversine formula serves the purposes of this problem very satisfactorily: