The position of an object in XY coordinates is converted to longitude and latitude to get a better and clear idea about the spot of the object on the surface of the earth. The position of an object can be expressed in a number of formats like the Military Grid Reference System (MGRS), Universal Transverse Mercator (UTM) system, geographic coordinate system that is latitude and longitude, and Universal Polar Stereographic (UPS). The geographic coordinate system is commonly used as it is simple and easy to understand.

- Skill level:
- Moderate

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## Instructions

- 1
Ensure that x, y and z values are specified in Cartesian coordinate system. The formula used is derived with the assumption that x, y and z values are defined in Cartesian coordinate system.

- 2
Assign the values of the coordinates to x, y and z. Assume the value 6371km to the variable R, which is the approximate radius of earth. This value is the scientifically derived value for radius of the earth.

- 3
Calculate latitude and longitude using the formula: latitude = asin (z/R) and longitude = atan2 (y,x). In this formula, we have the values of x, y, z and R from step 2. Asin is arc sin, which is a mathematical function, and atan2 is a variation of the arc tangent function. The symbol * stands for multiplication. The above two formulas are derived from the following formulas: x = R * cos (latitude) * cos (longitude); y = R * cos (latitude) * sin (longitude); z = R * sin (latitude). In this formula, sin and cos are mathematical functions. Asin and atan value can be calculated using a trigonometry calculator. The value of atan2 can be calculated using the formula a atan2 (y, x) = 2 atan (y/√(x²+y²)-x). Here √ indicates square root, here square root of (x²+y²).

#### Tips and warnings

- All the values should be in the same unit system. That is, if the trigonometric functions expect radians, then latitude and longitude should also be in radians.