# Trigonometry Involved in GPS

By John Papiewski i Jupiterimages/Creatas/Getty Images

Global Positioning System technology feeds electronic navigation devices with location data that helps guide aircraft, ships, vehicles or pedestrians toward their destinations. GPS uses some fairly complex calculations, largely based on the trigonometry surveyors use. Satellites in space transmit precisely timed signals to the GPS receiver, which determines latitude, longitude and altitude to within a few yards.

## Satellites

The GPS system uses 24 satellites in Earth orbit, each transmitting a unique coded signal to an earthbound receiver. Each satellite has an atomic clock which measures time accurately to 8 billionths of a second per day, according to GPS.gov. To get a proper location, the receiver must receive direct signals from four different satellites at the same time. The imaginary line to a satellite from the GPS unit and between each satellite forms the sides of several triangles which the receiver uses for trigonometric calculations.

## Time and Distance

To use trigonometry to determine location, you need the length of at least one of the triangle's sides. A GPS device does this by calculating the time it takes for the satellite signal to reach it. Because the speed of radio signals is the same as the speed of light, the unit accurately determines the distance to one satellite by multiplying the signal's travel time by the speed of light.

## Law of Cosines

A trigonometric rule called the Law of Cosines allows the GPS receiver to calculate its distance from each satellite. The Law of Cosines applies to GPS technology as follows:

d^2 = Re^2 + Rs^2 + 2ReRs*Cos(L)

Here, "d" is the distance from the satellite to the receiver, "Re" is the radius of the Earth, "Rs" is the radius of the satellite's orbit, and "L" is the angle formed between the straight lines from the center of the Earth to the satellite and from the center of the Earth to the GPS receiver.

## Intersecting Spheres

The distance to one satellite locates the GPS receiver inside an imaginary sphere whose radius is the distance. A second satellite narrows this down to the circle formed where two spheres intersect. The distance from three satellites produces three spheres which intersect at a point. A fourth satellite establishes the location of the GPS receiver on the Earth, along with the device's altitude.