I am trying to check my work on calculating the angle between two sats and a ground station observing those two sats. I have the az and elevation angles to each sat from the ground station. What would be the equation to get the angle Sat1-GS-Sat2?

I know that it uses the law of Cosines, but not sure if I was using the correct equation. Here is what I am using:

Angle = acos((abs(sin("V_Sat_1_EL")*sin("V_Sat_2_EL")))+(cos("V_Sat_1_EL")*cos("V_Sat_2_EL")*cos(abs("V_Sat_2_AZ"-"V_Sat_1_AZ"))))



2 Answers 2


This figure from Wikipedia:Spherical law of cosines applies if the ground station is at the center of the sphere, u is the ground station's zenith, and v and w are the apparent positions of the satellites as seen from the ground station.

spherical triangle diagram

Then a and b are the satellites' zenith angles, i.e. 90° minus their elevation angles. C is the difference between the satellites' azimuths, and c is the angle you want.

Using the cosine rule for sides $$\cos c = \cos a \cos b + \sin a \sin b \cos C$$ and the rules for sines and cosines of complementary angles, the expression in the question looks correct if the absolute value operations are omitted.


If you only need angles, you need interior angle triangle forumula, all angles in interior sum is equal to 180° . only you need Angle Sat 1 + Angle Sat 2 + Angle Sat 1-2 = 180°, if you need distances A, B , C you need use polar coordenates. lat, long and elevation.

  • $\begingroup$ I don't know the angle between the two satellites, only the AZ and EL to each from the GS. Think of the GS as the center of a sphere, and the two satellites are points on that sphere and I have two angles from the center of the sphere that give the position of where each Satellite is on the surface of the sphere. I need the angle that is formed by that GS looking at each Satellite. $\endgroup$
    – Shaps
    Nov 4, 2019 at 16:54
  • $\begingroup$ AZ is Azimuth , and EL is Elevation ? because Azimuth is a angular meassure $\endgroup$
    – Adrian R
    Nov 4, 2019 at 17:43
  • $\begingroup$ I should clarify. I know the AZ (angle) and Elevation Angle to each. So imagine my Ground Station is sitting at the center. My AZ to one satellite is the angle I rotate myself to look at it and then my elevation is how high I need to look at it. I have both of those for each Satellite. I need to figure out the angle that is formed between the first satellite, the Ground Station, and then the second satellite. I hope that this makes sense. $\endgroup$
    – Shaps
    Nov 4, 2019 at 21:44
  • 2
    $\begingroup$ This answer is for Euclidian planar geometry but the question is about spherical trigonometry; triangles drawn on the surface of a unit sphere. $\endgroup$
    – uhoh
    Nov 5, 2019 at 1:08

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