Spherical Law of Cosines

Suppose that we want to find the five nearest places to (47.470779, -87.890699) using Spherical Law of Cosines, the following MySQL syntax would easily accomplish it:

SELECT * AS place, 
  (DEGREES(
    ACOS(
      SIN(RADIANS(47.470779)) * SIN(RADIANS(geo_latitude)) + 
      COS(RADIANS(47.470779)) * COS(RADIANS(geo_latitude)) * 
      COS(RADIANS(-87.890699 - geo_longitude))
    ) 
  ) * 60 * 1.1515) 
AS distance FROM `MyDatabase`.`allplaces` ORDER BY distance ASC LIMIT 20 ;

Haversine Formula

Suppose that we want to find the five nearest places to (47.470779, -87.890699) using Haversine Formula, the following MySQL syntax would easily accomplish it [please refer here to read about the motivation/reasoning behind using atan2() function instead of sin(min()).

SELECT * , 
      ((2 * 3960 * 
        ATAN2(
          SQRT(
            POWER(SIN((RADIANS(47.470779 - geo_latitude))/2), 2) +
            COS(RADIANS(geo_latitude)) *
            COS(RADIANS(47.470779 )) *
            POWER(SIN((RADIANS(-87.890699 - geo_longitude))/2), 2)
          ),
          SQRT(1-(
            POWER(SIN((RADIANS(47.470779 - geo_latitude))/2), 2) +
            COS(RADIANS(geo_latitude)) *
            COS(RADIANS(47.470779 )) *
            POWER(SIN((RADIANS(-87.890699 - geo_longitude))/2), 2)
          ))
        )
      )) AS distance  FROM `MyDatabase`.`allPlaces` ORDER BY distance LIMIT 20;