Work out an approximate lat/lng translation by a mileage distance

Work out an approximate lat/lng translation by a mileage distance

Trying to find the lat/lng coordinate that lies 10 miles north, south,
east and west of an origin coordinate, I have an approximate rule that
states:
Latitude: 1 deg = 110.54 km
Longitude: 1 deg = 111.320*cos(latitude) km
How might I write formulas such that I can pass them an origin coordinate
and a mileage and have each return to me the coordinate that lies 10 miles
either north, south, east and west of the origin coordinate?
Thanks