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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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.