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Geosynchronous satellites orbit at an altitude of 35,786 km, directly above the Equator, and assume that earth is a perfect sphere. Please any hints or detail computation would be really appreciated. Thanks

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You can simplify the problem by thinking in terms of the equatorial plane: represent the equator as the circle $x^2+y^2=R_e^2$ where $R_e$ is the equatorial radius of the Earth. Then put a satellite at $(d+R_e,0)$, where $d$ is the height above the surface. Now draw a line from this point which is tangent to the circle at the point of intersection. What is the point of intersection? –  dls Mar 7 '12 at 5:27

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There is a maximum latitude from which geosynchronous satellites are visible (assuming zero elevation above sea level). Therefore no number of geosynchronous satellites can ever cover the whole earth's surface.

If all you care about is relatively low latitudes, where most of the population of the earth lives, then two satellites could cover almost all the populated land mass, and three would be almost as good as an infinite number of satellites.

GPS satellites are not geosynchronous, but for applications like GPS, you need at least three satellites in the sky from a given location.

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On the number of sattelites required for GPS, see this question. –  joriki Mar 7 '12 at 13:52
    
Note that GPS satellites are not geosynchronous. (Edited my answer to reflect this.) –  Ben Crowell Jul 24 '12 at 0:21
    
Aside from the poles, each satellite sees a bit less than half the equator. Arthur C. Clarke's original paper lakdiva.org/clarke/1945ww points out that it takes three. –  Ross Millikan Jul 24 '12 at 1:41

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