Calculate second point knowing the starting point and distance

It seems you are measuring distance (R) in meters, and bearing (theta) counterclockwise from due east. And for your purposes (hundereds of meters), plane geometry should be accurate enough. In that case,

dx = R*cos(theta) ; theta measured counterclockwise from due east
dy = R*sin(theta) ; dx, dy same units as R

If theta is measured clockwise from due north (for example, compass bearings),
the calculation for dx and dy is slightly different:

dx = R*sin(theta)  ; theta measured clockwise from due north
dy = R*cos(theta)  ; dx, dy same units as R

In either case, the change in degrees longitude and latitude is:

delta_longitude = dx/(111320*cos(latitude))  ; dx, dy in meters
delta_latitude = dy/110540                   ; result in degrees long/lat

The difference between the constants 110540 and 111320 is due to the earth’s oblateness
(polar and equatorial circumferences are different).

Here’s a worked example, using the parameters from a later question of yours:

Given a start location at longitude -87.62788 degrees, latitude 41.88592 degrees,
find the coordinates of the point 500 meters northwest from the start location.

If we’re measuring angles counterclockwise from due east, “northwest” corresponds
to theta=135 degrees. R is 500 meters.

dx = R*cos(theta) 
   = 500 * cos(135 deg) 
   = -353.55 meters

dy = R*sin(theta) 
   = 500 * sin(135 deg) 
   = +353.55 meters

delta_longitude = dx/(111320*cos(latitude)) 
                = -353.55/(111320*cos(41.88592 deg))
                = -.004266 deg (approx -15.36 arcsec)

delta_latitude = dy/110540
               = 353.55/110540
               =  .003198 deg (approx 11.51 arcsec)

Final longitude = start_longitude + delta_longitude
                = -87.62788 - .004266
                = -87.632146

Final latitude = start_latitude + delta_latitude
               = 41.88592 + .003198
               = 41.889118

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