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A position on the surface of the earth is located by means of latitude and longitude, where latitude is measured from the equator to the position. Similarly, the position of astronomical objects is determined by their declination which is also measured from the equator.

A navigational triangle is determined by three points, two locations and the North or South pole, so two of the sides of the triangle are measured from the pole to the location and are therefore the complements of the latitude or the declination. Rather than calculate the complements, we make use of the following relationships to change the equations.

sin ( 90-x ) = cos x        cos ( 90-x ) = sin x

When sailing from point one (L1, LO1) to point two (L2, LO2) DLO = the difference in longitude. (degrees) C1 = course angle at point one. D = the distance from point one to point two. (degrees) cos D = sin L1 * sin L2 + cos L1 * cos L2 * cos DLO cos C1 = (sin L2 - sin L1 * cos D ) / ( cos L1 * sin D ) or sin C1 = ( cos L2 * sin DLO ) / sin D To convert D from degrees to nautical miles, multiply D by 60

The last equation will give ambiguous results (if you use it) so you must check to determine which of the two possible answers is the correct one.

The above method does not work out very well with a logarithmic solution, so early in the twentieth century a cosine-haversine formula was used for a solution by logarithms, requiring a table of haversines and log haversines.

Since that time better methods have been found, and so I believe that the cosine-haversine method is obsolete.

The law of cosine method may be the best for use in a computer program, or a programmable calculator.

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