How to calculate spherical trigonometry

Updated April 17, 2017

Spherical trigonometry is a area of geometry that deals with the triangles on the surface of a sphere. This branch of trigonometry developed in the early eighth century and has been used in navigation, stellar mapping, making geographic maps and improving sundials.

Determine the distance between two cities using spherical trigonometry. If you know the latitude and longitudes of the two cities, you can determine the distance between the two using the Cosine Rule (see Tips below), since the shortest distance between the two points is an arc on the great circle. The longitudes can be translated into distances based on the radius of the earth.

Find the heading or direction between two points by using the Sine Rule (see Tips below) of spherical trigonometry. A pilot can use the longitude and latitudes of two airports (one is the departure airport, the second is the arrival airport) and calculate what heading he should take to get to his destination by plugging the location values into the Sine Rule.

Calculate the declination or latitude of stars using spherical trigonometry. The declination of a star is its position above or below the celestial equator. Once a stars position is calculated it can be plotted on a star map or tabulated for navigational purposes. This done by using the following declination formula based on spherical trigonometry: Sin(d) = Sin(l)Cos(z) - Cos(l)Sin(z)Cos(a) Where "a" is the azimuth or apparent angle of an object in the sky; "z" is the zenith distance or the distance from the Earth's centre to a point in the sky; and "l" is the latitude of the location where the observation is being made.

Figure out the suns declination to determine sunset and sunrise positions. The suns declination changes throughout the year, on equinox the sun's declination is at 0 degrees; rising at azimuth 90 degrees in the east and setting at west at azimuth 270 degrees. Spherical Trigonometry can used to calculate exactly where the sun will rise on a given day. This calculated by the following formula: Cos(a) = Sin(d) / Cos(l) Where "a" is the azimuth or apparent angle of an object in the sky; "l" is the latitude of the location where the observation is being made; and "d" is the declination of the sun.

Make an accurate sundial. Initially sundials were made with a straight stick placed vertically in the ground, but this was not accurate because of the sun's position or declination changed at different times of the year. The sundials therefore required a different scale to be used during different times of the year. The problem was solved using spherical geometry: the sundial gnomon was given an angle. The angle is calculated using spherical trigonometry.


Unlike the regular triangle, the sum of the angles on a sphere is greater than 180, and the lengths "a," "b" and "c" represent arcs on a larger circle also known as a great circle. There are cosine and sine rules that are specifically used to develop formulas for spherical triangles: Cos(a) = cos(b) x cos(c) + sin(b) x sin(c) x cos(A) Cos(b) = cos(a) x cos(c) + sin(a) x sin(c) x cos(B) Using the Cosine Rule, you can determine the length of one side of the arc if you already know the length of two sides and the angle opposite the arc. Sin(a)/Sin(A) = Sin(b)/Sin(B) = Sin(c)/Sin(C) This rule can be used to find an angle if the length of two sides and one angle is known, or one length of two angles and only one side is known.

Things You'll Need

  • Calculator with trigonometry functions
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