How do I calculate spherical coordinates of a point "in" or "on" Earth from the entry point (lat/long), angle/direction of entry and distance/depth?

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All kids at some point ask themselves "if I dig a hole in the Earth and keep digging, where will I end up?". I certainly remember thinking this. I would like to try and work this out. But I have no idea where to start.

So if I wanted to figure out where in the world I'd be (lat/long/depth) if I dug a hole (1km) at an angle of 25 degrees, in the direction of north, from my current GPS coords, how could I calculate this?

A weird question, I know. But it's fascinated me for a while so I thought it was worth asking.