Trigonometry on the Ellipsoid Mystery Cache

This geocache cannot be found at the posted coordinates.  You need to solve the following puzzle to find this geocache.

This problem needs to be solved on the WGS84 ellipsoid.   You are provided with the following two points:

Point A N37 43.720 W121 43.395
Point B N37 39.936 W121 43.395

The geocache is located at point C.  Point C is 2km from point A.   The bearing from point C to point B is 166.286726274 degrees from true north.   Point C is south of Point A.   While this puzzle is similar to Trigonometry on the Plane it must be solved in a completely different manner because the problem space is the WGS84 ellipsoid and not a plane.

Programming skills are not required to solve this.   Some manual iteration and a little patience with a tool such as Fizzycalc can provide a solution. If you use fizzycalc you'll need to find the cache location such that your bearing from cache to B matches to three decimal places, ie 166.286xxx

You can validate your solution with Certitude

You can validate your puzzle solution with certitude.

Please only search for this cache during daylight hours.